text:
\section{Introduction} \label{sec:intro}

A fundamental problem in spectral graph theory is the classification and characterization of graphs with bounded eigenvalues. When we talk about eigenvalues of a graph we always refer to its adjacency matrix. In this paper, we study the families of graphs with eigenvalues bounded from below. Let $\G(\la)$ be the family of graphs with smallest eigenvalue at least $-\lambda$. For the sake of comparison, we mention the family $\G'(\la)$ of graphs with spectral radius (or, equivalently, largest eigenvalue) at most $\lambda$.

\begin{remark}
    Since we rarely work with subgraphs that are not induced, all subgraphs are induced throughout, and we refer to subgraphs that are not necessarily induced by \emph{general subgraphs}.
\end{remark}

The Cauchy interlacing theorem implies that both $\G'(\la)$ and $\G(\la)$ are closed under taking subgraphs. The natural question arises whether it is possible to define each of these families by a finite set of forbidden subgraphs.

\begin{definition}
    Given a family $\mathcal{G}$ of graphs, a family $\mathcal{F}$ of graphs is a \emph{forbidden subgraph characterization} for $\mathcal{G}$ if the family $\mathcal{G}$ consists exactly of graphs that do not contain any member of $\mathcal{F}$ as a subgraph, and a graph $F$ is a \emph{minimal forbidden subgraph} for $\mathcal{G}$ if $F$ itself is not in $\mathcal{G}$ but every proper subgraph of $F$ is in $\mathcal{G}$.
\end{definition}

Note that the most economical forbidden subgraph characterization for $\mathcal{G}$ consists precisely of the minimal forbidden subgraphs for $\mathcal{G}$. Thus the existence of a finite forbidden subgraph characterization for $\mathcal{G}$ is equivalent to the finiteness of the minimal forbidden subgraphs for $\mathcal{G}$. In 1992 Bussemaker and Neumaier made the following remark in \cite[p~599]{BN92}:

\begin{quote}
    It would be interesting to know the set of numbers $m$, $-m$ such that $\mathcal{G}_m^\#$ [the set of minimal forbidden subgraphs for $\mathcal{G}'(m)$] or $\mathcal{G}_{-m}^\#$ [the set of minimal forbidden subgraphs for $\mathcal{G}(m)$] are finite; however, these seem to be very difficult problems.
\end{quote}

\begin{figure}
    \centering
    \begin{tikzpicture}[very thick, scale=0.4, baseline=(v.base)]
        \coordinate (v) at (0,0);
        \draw[black] (0,0) -- (4,2) node[vertex]{} -- (8,0);
        \draw[black, loosely dotted] (2,0) -- (6,0);
        \draw[black] (0,0) node[vertex]{} -- (2,0) node[vertex]{};
        \draw[black] (6,0) node[vertex]{} -- (8,0) node[vertex]{};
        \node at (4,-1.5) {$\widetilde{A}_n$ ($n \ge 2$)};
    \end{tikzpicture}\qquad
    \begin{tikzpicture}[very thick, scale=0.4, baseline=(v.base)]
        \coordinate (v) at (0,0);
        \draw[black] (2,0) -- (2,2) node[vertex]{};
        \draw[black] (10,0) -- (10,2) node[vertex]{};
        \draw[black, loosely dotted] (4,0) -- (8,0);
        \draw[black] (0,0) node[vertex]{} -- (2,0) node[vertex]{} -- (4,0) node[vertex]{};
        \draw[black] (8,0) node[vertex]{} -- (10,0) node[vertex]{} -- (12,0) node[vertex]{};
        \node at (6,-1.5) {$\widetilde{D}_n$ ($n \ge 4$)};
    \end{tikzpicture}\\
    \begin{tikzpicture}[very thick, scale=0.4, baseline=(v.base)]
        \coordinate (v) at (0,0);
        \draw[black] (4,0) -- (4,2) node[vertex]{} -- (6,2) node[vertex]{};
        \draw[black] (0,0) node[vertex]{} -- (2,0) node[vertex]{} -- (4,0) node[vertex]{} -- (6,0) node[vertex]{} -- (8,0) node[vertex]{};
        \node at (4,-1.5) {$\widetilde{E}_6$};
    \end{tikzpicture}\qquad
    \begin{tikzpicture}[very thick, scale=0.4, baseline=(v.base)]
        \coordinate (v) at (0,0);
        \draw[black] (6,0) -- (6,2) node[vertex]{};
        \draw[black] (0,0) node[vertex]{} -- (2,0) node[vertex]{} -- (4,0) node[vertex]{} -- (6,0) node[vertex]{} -- (8,0) node[vertex]{} -- (10,0) node[vertex]{} -- (12,0) node[vertex]{};
        \node at (6,-1.5) {$\widetilde{E}_7$};
    \end{tikzpicture}\qquad
    \begin{tikzpicture}[very thick, scale=0.4, baseline=(v.base)]
        \coordinate (v) at (0,0);
        \draw[black] (4,0) -- (4,2) node[vertex]{};
        \draw[black] (0,0) node[vertex]{} -- (2,0) node[vertex]{} -- (4,0) node[vertex]{} -- (6,0) node[vertex]{} -- (8,0) node[vertex]{} -- (10,0) node[vertex]{} -- (12,0) node[vertex]{} -- (14,0) node[vertex]{};
        \node at (7,-1.5) {$\widetilde{E}_8$};
    \end{tikzpicture}
    \caption{Maximal connected graphs with spectral radius at most $2$. The number of vertices is one more than the given index. In particular, $\widetilde{D}_4$ is actually a star with four leaves.} \label{fig:dynkin}
\end{figure}

The specific families $\mathcal{G}'(2)$ and $\mathcal{G}(2)$ are well understood. One of the earliest results dates back to 1970 when Smith~\cite{S70} determined all the connected graphs in $\mathcal{G}'(2)$ --- they are general subgraphs of the extended Dynkin diagrams in \cref{fig:dynkin}. The family $\mathcal{G}(2)$ is much richer --- it contains not only all the graphs in $\mathcal{G}'(2)$, but also all the line graphs.\footnote{A line graph $L(H)$ of a graph $H$ is obtained by creating a vertex per edge in $H$, and connecting two vertices if and only if the corresponding edges in $H$ have a vertex in common. The adjacency matrix of $L(H)$ can be written as $B^TB - 2I$, where $B$ is the vertex-edge incidence matrix of $H$, hence the smallest eigenvalue of $L(H)$ is at least $-2$.\label{footnote:line-graph}} The classification of $\mathcal{G}(2)$ culminated in a beautiful theorem of Cameron, Goethals, Seidel, and Shult~\cite{CGSS76} who related $\mathcal{G}(2)$ to root systems from the representation theory of semisimple Lie algebras (see \cref{thm:cgss}). We refer the reader to the monograph \cite{CRS04} for a comprehensive account of $\mathcal{G}(2)$.

These classification theorems can be used to establish quantitative answers to Bussemaker and Neumaier's problems. For $\mathcal{G}'(2)$, Cvetkovi\'{c}, Doob and Gutman~\cite[Theorem~2.8]{CDG82} determined that there are $18$ minimal forbidden subgraphs. For $\mathcal{G}(2)$, Rao, Singhi and Vijayan~\cite[Theorem~4.1]{RSV81} observed that the number of vertices in any minimal forbidden subgraph is at most $37$, which was eventually perfected to $10$ by Kumar, Rao and Singhi~\cite{KRS82}. A computer search by Bussemaker and Neumaier~\cite[p~596]{BN92} established that there are, in total, $1812$ minimal forbidden subgraphs for $\mathcal{G}(2)$.

In a recent work, the authors of the current paper resolved the first problem of Bussemaker and Neumaier.\footnote{The original statement of \cite[Theorem~1]{JP20} determines the set of $\lambda$ for which the family $\G'(\la)$ can be defined by a finite set of forbidden \emph{general} subgraphs. Using the well known fact that the spectral radius of $G_1$ is at most that of $G_2$ whenever $G_1$ is a general subgraph of $G_2$, one can show that if $\mathcal{G}$ is a forbidden general subgraph characterization for $\G'(\la)$, then the family $\dset{H}{\exists\, G \in \mathcal{G} \text{ s.t. }G\text{ is a general subgraph of }H\text{ on the same vertex set}}$ is a forbidden (induced) subgraph characterization for $\G'(\la)$, and so the original statement of \cite[Theorem~1]{JP20} is equivalent to \cref{thm:forb1}.}

\begin{theorem}[Theorem~1 of Jiang and Polyanskii~\cite{JP20}] \label{thm:forb1}
    For every integer $m \ge 2$, let $\beta_m$ be the largest root of $x^{m+1} = 1 + x + \dots + x^{m-1}$, and let $\alpha_m := \beta_m^{1/2} + \beta_m^{-1/2}$. The family $\G'(\la)$ of graphs with spectral radius at most $\lambda$ has a finite forbidden subgraph characterization if and only if $\lambda < \lambda'$ and $\lambda \not\in\sset{\alpha_2, \alpha_3, \dots}$, where
    \[
        \lambda' := \lim_{m\to\infty} \alpha_m = \sqrt{2+\sqrt5} \approx 2.05817.
    \]
\end{theorem}

The first main theorem of this paper resolves the remaining problem of Bussemaker and Neumaier --- $\G(\la)$ enjoys a straightforward threshold phenomenon.

\begin{theorem} \label{thm:main1}
    The family $\G(\la)$ of graphs with smallest eigenvalue at least $-\lambda$ has a finite forbidden subgraph characterization if and only if $\lambda < \lambda^*$, where
    \[
        \lambda^* := \beta^{1/2} + \beta^{-1/2} \approx 2.01980,
    \]
    and $\beta$ is the unique real root of $x^3 = 1 + x$.\footnote{The exact value of $\beta$, known as the plastic number, is $\sqrt[3]{(9+\sqrt{69})/18} + \sqrt[3]{(9-\sqrt{69})/18}$.}
\end{theorem}

\begin{remark}
    The constants defined in \cref{thm:forb1,thm:main1} satisfy $2 < \lambda^* = \alpha_2 < \alpha_3 < \dots < \lambda'$.
\end{remark}

As a byproduct, we determine all the limit points of the set of smallest graph eigenvalues of graphs. Let $\Lambda_1$ consist of $\lambda \in \mathbb{R}$ such that $-\lambda$ is the smallest eigenvalue of some graph. Here we invert the smallest eigenvalues of graphs similarly to how we define $\G(\la)$. Before our work, Hoffman~\cite{H77a} characterized all the limit points of $\Lambda_1$ in $(-\infty, 2]$. His result involves a technical qualification, which was conjectured to be always true, and was later established by Greaves et al.~\cite[Theorem~1]{GKMST15}. Doob~\cite[Theorem~9]{D89} observed that every real number in $\sset{\alpha_2, \alpha_3, \dots} \cup [\lambda',\infty)$ is a limit point of $\Lambda_1$, and conjectured that $\alpha_2, \alpha_3, \dots$ are the only limit points of $\Lambda_1$ in $(2, \lambda')$. We refute this conjecture by finding all the limit points of $\Lambda_1$ in $(2, \lambda')$.

\begin{corollary} \label{cor:main1}
    For every $\lambda > 2$, $-\lambda$ is a limit point of the set of smallest eigenvalues of graphs if and only if $\lambda \ge \lambda^*$.
\end{corollary}

We next turn our attention to \emph{signed graphs}, which are graphs whose edges are each labeled by $+$ or $-$. Throughout the paper we decorate variables for signed graphs with the $\pm$ superscript. When we talk about eigenvalues of a signed graph $G^\pm$ on $n$ vertices, we refer to its \emph{signed adjacency matrix} --- the $n \times n$ matrix whose $(i,j)$-th entry is $1$ if $ij$ is a positive edge, and $-1$ if $ij$ is a negative edge, and $0$ otherwise.

It still makes sense to speak of forbidden subgraph characterization for families of signed graphs with eigenvalues bounded from below. Our second main theorem establishes the same threshold phenomenon for the family of signed graphs with smallest eigenvalue at least $-\lambda$.

\begin{theorem} \label{thm:main2}
    The family $\G^\pm(\la)$ of signed graphs with smallest eigenvalue at least $-\lambda$ has a finite forbidden subgraph characterization if and only if $\lambda < \lambda^*$.
\end{theorem}

Notice that if $\mathcal{F}^\pm$ is a finite forbidden subgraph characterization for $\G^\pm(\la)$, then the set of all-positive signed graphs in $\mathcal{F}^\pm$ is a finite forbidden subgraph characterization for $\G(\la)$. Therefore, for $\lambda < \lambda^*$, \cref{thm:main2} implies \cref{thm:main1}. However, in \cref{sec:forb}, we still provide the complete proof of \cref{thm:main1} to illustrate the key ideas.

As an application we prove the following result on symmetric matrices whose entries are all integers.

\begin{corollary} \label{cor:mat}
    For every $\lambda < \lambda^*$, there exists $n \in \mathbb{N}$ such that for every symmetric integer matrix $A$ whose diagonal entries are all zero, if $A$ has an eigenvalue less than $-\lambda$, then so does one of its principal submatrix of order at most $n$. However, the same conclusion does not hold for any $\lambda \ge \lambda^*$.
\end{corollary}

\cref{cor:mat} is inspired by Vijayakumar~\cite{V87}. He first showed that every minimal forbidden subgraph for $\mathcal{G}^\pm(2)$ has at most $10$ vertices, and then he proved, as an application, a quantitative version of \cref{cor:mat} in a special case ---  for $\lambda = 2$ the natural number $n$ can be as small as (but no smaller than) $10$.

Finally we turn our attention to the largest eigenvalue of a signed graph. We denote the eigenvalues of a signed graph $G^\pm$ by $\lambda_1(G^\pm) \le \lambda_2(G^\pm) \le \dots $ in ascending order, and by $\lambda^1(G^\pm) \ge \lambda^2(G^\pm) \ge \dots$ in descending order. Observing that for every signed graph $G^\pm$, $\lambda_1(G^\pm) \ge -\lambda$ if and only if $\lambda^1(-G^\pm) \le \lambda$, where $-G^\pm$ negates all the edge signs of $G^\pm$, we obtain an immediate consequence of \cref{thm:main2}.

\begin{corollary} \label{cor:main2}
    The family $\G^\mp(\la)$ of signed graphs with largest eigenvalue at most $\lambda$ has a finite forbidden subgraph characterization if and only if $\lambda < \lambda^*$. \qed
\end{corollary}

Our motivation to understand the forbidden subgraph characterization for $\G^\mp(\la)$ comes from the problem of determining the maximum size of a \emph{spherical two-distance set} with two fixed angles (one acute and one obtuse) in high dimensions. For fixed $-1 \le \beta < 0 \le \alpha < 1$, let $N_{\al,\be}(d)$ denote the maximum number of unit vectors in $\mathbb{R}^d$ where all pairwise inner products lie in $\sset{\alpha,\beta}$.

In the special case $\alpha = -\beta$, which corresponds to \emph{equiangular lines}, recent work \cite{B16,BDSK18,JP20} culminated in a solution \cite{JTYZZ19} of Jiang, Tidor, Yao, Zhang and Zhao to the problem of determining $N_{\alpha, -\alpha}(d)$ for sufficiently large $d$. We refer the reader to \cite[Section~1]{JTYZZ19} for earlier developments on equiangular lines with fixed angles in high dimensions.

For the general case $-1 \le \beta < 0 \le \alpha < 1$, in their subsequent work \cite{JTYZZ20}, Jiang et al.\ proposed a conjecture on the limit of $N_{\al,\be}(d)/d$ as $d\to\infty$. To state their conjecture, we need the following spectral graph quantity.

\begin{definition}
    Given $\lambda > 0$ and $p \in \mathbb{N}$, define the parameter
    \[
        k_p(\la) = \inf\dset{\frac{\abs{G^\pm}}{\operatorname{mult}(\lambda, G^\pm)}}{\chi(G^\pm) \le p \text{ and }\lambda^1(G^\pm) = \lambda},
    \]
    where $\abs{G^\pm}$ is the number of vertices of $G^\pm$, $\operatorname{mult}(\lambda, G^\pm)$ is the multiplicity of $\lambda$ as an eigenvalue of $G^\pm$, and $\chi(G^\pm)$ is the chromatic number of $G^\pm$.
\end{definition}

We postpone the definition of $\chi(G^\pm)$, which takes values in $\mathbb{N}^+ \cup \sset{\infty}$, to \cref{subsec:two-distance}. We now state the conjecture on $N_{\al,\be}(d)$.

\begin{conjecture}[Conjecture~1.7 of Jiang et al.~\cite{JTYZZ20}] \label{conj:main}
    Fix $-1 \le \beta < 0 \le \alpha < 1$. Set $\lambda = (1-\alpha)/(\alpha - \beta)$ and $p = \lfloor -\alpha/\beta \rfloor + 1$. Then
    \[
        N_{\al,\be}(d) = \begin{dcases}
            \frac{k_p(\la) d}{k_p(\la)-1} + o(d) & \text{if }k_p(\la) < \infty, \\
            d + o(d) & \text{otherwise}.
        \end{dcases}
    \]
\end{conjecture}

\cref{conj:main} was confirmed in \cite{JTYZZ20} for $p \le 2$ or $\lambda \in \sset{1,\sqrt2, \sqrt3}$. Building on the framework developed there, we establish \cref{conj:main} for all $\lambda < \lambda^*$ as an application of \cref{cor:main2}.

The rest of the paper is organized as follows. We prove \cref{thm:main1,cor:main1} in \cref{sec:forb}, and we prove \cref{thm:main2} in \cref{sec:forb-signed}. Part of the proofs are computer-assisted with validated numerics. In \cref{sec:numerics} we explain our computer-aided proof and how anyone can recreate it independently. In \cref{sec:app} we give applications of \cref{thm:main2} to symmetric integer matrices and spherical two-distance sets. In \cref{sec:discussion}, we discuss open problems related to the classification of graphs in $\mathcal{G}(\lambda^*) \setminus \mathcal{G}(2)$, and a possible way to establish more instances of \cref{conj:main}.

\section{Forbidden subgraphs for $\G(\la)$} \label{sec:forb}

We break the proof of \cref{thm:main1} into three cases $\lambda < 2$, $\lambda \in [2, \lambda^*)$ and $\lambda \ge \lambda^*$, where $\lambda^* \approx 2.01980$ is defined as in \cref{thm:main1}. It is worth pointing out the spectral graph interpretation of the peculiar constant $\lambda^*$.

\begin{figure}
    \centering
    \begin{tikzpicture}[very thick, scale=0.3, baseline=(v.base)]
        \draw (3,0) -- +(0,2) node[vertex]{};
        \draw (-1,0) node[vertex]{} -- (1,0) node[vertex]{} -- (3,0) node[vertex]{} -- (5,0) node[vertex]{} -- (7,0) node[vertex]{} -- (9,0) node[vertex]{} -- (11,0) node[vertex]{} -- (13,0) node[vertex]{} -- (15,0) node[vertex]{};
        \draw (-2, 0) node[left] (v) {$E_{2,n} = $};
        \draw [braket] (15,0.1) -- (5,0.1) node [black,midway,yshift=-15pt] {\footnotesize $n$};
    \end{tikzpicture}
    \caption{The graph $E_{2,n}$ has $n+4$ vertices.} \label{fig:e2n}
\end{figure}

\begin{proposition}[{\cite[Lemma~5(e)]{JP20}}] \label{lem:e2n}
    The graphs $E_{2,n}$ defined in \cref{fig:e2n} satisfy that, as $n \to \infty$, $\lambda^1(E_{2,n})$ increases to $\lambda^*$ or equivalently $\lambda_1(E_{2,n})$ decreases to $-\lambda^*$.\footnote{The equivalence is due to the fact that the spectrum of a bipartite graph is symmetric with respect to $0$.} \qed
\end{proposition}

\subsection{Proof of \cref{thm:main1} for $\lambda < 2$} \label{subsec:less-than-2}

Hoffman \cite{H77a} demonstrated several sequences of graphs $G_1, G_2, \dots$ such that $G_n$ is a subgraph of $G_{n+1}$ for every $n$ and $\lim \lambda_1(G_n) \le -2$. To state these results, we introduce the following notions.

\begin{definition} \label{def:extension}
    Given a nonempty vertex subset $A$ of a graph $F$, $\ell \in \mathbb{N}$, and $m \in \mathbb{N}^+$,
    \begin{enumerate}[nosep, label=(\alph*)]
        \item the \emph{path extension} $(F, A, \ell)$ is obtained from $F$ by adding a path $v_0 \dots v_\ell$ of length $\ell$, and connecting $v_0$ to every vertex in $A$;\footnote{When $\ell = 0$, the path of length $\ell$ is simply a single vertex.}
        \item the \emph{path-clique extension} $(F, A, \ell, K_m)$ is further obtained from $(F, A, \ell)$ by adding a clique of size $m$, and connecting every vertex in the clique to $v_\ell$;
        \item the \emph{clique extension} $(F, A, K_m)$ is obtained from $F$ by adding a clique of size $m$, and connecting every vertex in the clique to every vertex in $A$.
    \end{enumerate}
    See \cref{fig:extensions} for schematic drawings of these extensions.
\end{definition}

\begin{figure}[b]
    \centering
    \begin{tikzpicture}[very thick, scale=0.5, baseline=(v.base)]
        \coordinate (v) at (0,0);
        \draw[rounded corners=14pt] (-7.8, -1) rectangle (-4, 1) {};
        \fill[litegray] (-3,0) -- (-4.68,0.733) -- (-4.68,-0.733) -- cycle;
        \fill[litegray] (-5, 0) circle (0.8);
        \draw (-5,0) node{$A$};
        \draw (-7,0) node{$F$};
        \draw (-3,0) node[vertex]{} -- (-2,0) node[vertex]{} -- (-1,0) node[vertex]{} -- (0,0) node[vertex]{} -- (1,0) node[vertex]{};
        \draw [braket] (1,0.2) -- (-3,0.2) node [black,midway,yshift=-15pt] {\footnotesize $\ell + 1$};
    \end{tikzpicture}\qquad%
    \begin{tikzpicture}[very thick, scale=0.5, baseline=(v.base)]
        \coordinate (v) at (0,0);
        \draw[rounded corners=14pt] (-7.8, -1) rectangle (-4, 1) {};
        \fill[litegray] (-3,0) -- (-4.68,0.733) -- (-4.68,-0.733) -- cycle;
        \fill[litegray] (1,0) -- (2.28,0.96) -- (2.28,-0.96) -- cycle;
        \fill[litegray] (-5, 0) circle (0.8);
        \fill[litegray] (3, 0) circle (1.2);
        \draw (-5,0) node{$A$};
        \draw (-7,0) node{$F$};
        \draw (3,0) node{$K_m$};
        \draw (-3,0) node[vertex]{} -- (-2,0) node[vertex]{} -- (-1,0) node[vertex]{} -- (0,0) node[vertex]{} -- (1,0) node[vertex]{};
        \draw [braket] (1,0.2) -- (-3,0.2) node [black,midway,yshift=-15pt] {\footnotesize $\ell + 1$};
    \end{tikzpicture}\qquad%
    \begin{tikzpicture}[very thick, scale=0.5, baseline=(v.base)]
        \coordinate (v) at (0,0);
        \draw[rounded corners=14pt] (-7.8, -1) rectangle (-4, 1) {};
        \fill[litegray] (-5.107,0.79) -- (-2.16,1.19) -- (-2.16,-1.19) -- (-5.107,-0.79) -- cycle;
        \fill[litegray] (-5, 0) circle (0.8);
        \fill[litegray] (-2, 0) circle (1.2);
        \draw (-5,0) node{$A$};
        \draw (-7,0) node{$F$};
        \draw (-2,0) node{$K_m$};
    \end{tikzpicture}
    \caption{Schematic drawings of the path extension $(F, A, \ell)$, the path-clique extension $(F, A, \ell, K_m)$ and the clique extension $(F, A, K_m)$.} \label{fig:extensions}
\end{figure}

We compile some of Hoffman's computation and two classical results in the following lemma.

\begin{lemma} \label{lem:hoffman}
    Denote by $C_n$ the cycle of length $n$, $V_2(C_n)$ a set of two adjacent vertices of $C_n$, $K_n$ the complete graphs with $n$ vertices, $\overline{K}_2$ the null graph with $2$ vertices, $P_n$ the path of length $n$, and $S_n$ the star with $n$ leaves. The smallest eigenvalues of these graphs or their extensions satisfy:
    \begin{itemize}[nosep]
        \item[(c1)] $\lim_m \lambda_1(C_n, V_2(C_n), \ell, K_m) = -2$ for fixed $n \ge 3$ and $\ell \in \mathbb{N}$; \wlabel{lem:c1}{c1}
        \item[(c2)] $\lim_m \lambda_1(C_n, V_2(C_n), K_m) \le -2$ for fixed $n \ge 3$; \wlabel{lem:c2}{c2}
        \item[(k)] $\lim_m \lambda_1(K_m, V(K_m), \ell, K_m) = -2$ for fixed $\ell \in \mathbb{N}$; \wlabel{lem:k}{k}
        \item[(n1)] $\lim_m \lambda_1(\overline{K}_2, V(\overline{K}_2), \ell, K_m) = -2$ for fixed $\ell \in \mathbb{N}$; \wlabel{lem:n1}{n1}
        \item[(n2)] $\lim_m \lambda_1(\overline{K}_2, V(\overline{K}_2), K_m) = -2$; \wlabel{lem:n2}{n2}
        \item[(p)] $\lim_n \lambda_1(P_n) = -2$; \wlabel{lem:p}{p}
        \item[(s)] $\lambda_1(S_n) = -\sqrt{n}$. \wlabel{lem:s}{s} \qed
    \end{itemize}
\end{lemma}

\begin{remark}
    In \cref{lem:hoffman}, (\ref{lem:c1}) to (\ref{lem:n2}) are taken directly from \cite[Lemmas~2.4 to 2.9]{H77a}, and (\ref{lem:p}) and (\ref{lem:s}) follow from the classical results $\lambda^1(P_n) = 2\cos(\pi/(n+2))$ and $\lambda^1(S_n) = \sqrt{n}$.
\end{remark}

\cref{lem:hoffman} allows us to build a finite set of forbidden subgraphs for $\G(\la)$. To state the result, we introduce the following definition.

\begin{definition}[Extension family]
    For every graph $F$ and $\ell, m \in \mathbb{N}^+$, the \emph{extension family} $\mathcal{X}(F, \ell, m)$ of $F$ consists of the path-extension $(F, A, \ell)$, the path-clique extensions $(F, A, \ell_0, K_m)$, where $\ell_0 \in \sset{0, \dots, \ell-1}$, and the clique extension $(F, A, K_m)$ for every nonempty vertex subset $A$ of $F$.
\end{definition}

\begin{lemma} \label{lem:hoffman-extension}
    For every $\lambda < 2$, there exist $\ell, m \in \mathbb{N}^+$ such that both the extension family $\mathcal{X}(C, \ell, m)$ of the claw graph $C$ and the extension family $\mathcal{X}(D, \ell, m)$ of the diamond graph $D$ are disjoint from $\G(\la)$.
\end{lemma}

\begin{figure}[t]
    \centering
    \begin{tikzpicture}[very thick, scale=0.5]
        \draw (0,0) -- (-1,1);
        \draw (0,0) -- (0,2);
        \draw (0,0) -- (1,1);
        \node[circ] at (0,0) {0};
        \node[circ] at (-1,1) {1};
        \node[circ] at (0,2) {2};
        \node[circ] at (1,1) {3};
        \draw (5,0) -- +(-1,1) -- +(0,2);
        \draw (5,0) -- +(1,1) -- +(0,2);
        \draw (5,0) -- +(0,2);
        \node[circ] at (5,0) {0};
        \node[circ] at (4,1) {1};
        \node[circ] at (5,2) {2};
        \node[circ] at (6,1) {3};
    \end{tikzpicture}
    \caption{The claw graph and the diamond graph.} \label{fig:claw-and-diamond}
\end{figure}

\begin{proof}
    From \cref{lem:hoffman}(\ref{lem:p}), we obtain $\ell \in \mathbb{N}^+$ such that $P_\ell \not\in \G(\la)$. Clearly, for every nonempty $A$, the path extensions $(C, A, \ell)$ and $(D, A, \ell)$, each of which contains $P_\ell$ as a subgraph, are not in $\G(\la)$. With hindsight, using \cref{lem:hoffman}(\ref{lem:c1}, \ref{lem:c2}, \ref{lem:n1}, \ref{lem:n2}), we choose $m \in \mathbb{N}^+$ such that none of the following graphs
    \begin{equation} \label{eqn:hoffman-extension}
        (C_3, V_2(C_3), \ell_0, K_m), (C_3, V_2(C_3), K_m), (\overline{K}_2, V(\overline{K}_2), \ell_0, K_m), (\overline{K}_2, V(\overline{K}_2), K_m),
    \end{equation}
    where $\ell_0 \in \sset{0, \dots, \ell + 1}$, is in $\G(\la)$.

    It suffices to prove that every clique extension and every path-clique extension in the extension families $\mathcal{X}(C, \ell, m)$ and $\mathcal{X}(D, \ell, m)$ contains one of the graphs in \eqref{eqn:hoffman-extension} as a subgraph. Label the vertices of $C$ and $D$ as in \cref{fig:claw-and-diamond}, and pick an arbitrary $\ell_0 \in \sset{0, \dots, \ell - 1}$. The clique extension $(C, A, K_m)$ and the path-clique extension $(C, A, \ell_0, K_m)$ respectively contain
    \begin{align*}
        (\overline{K}_2, V(\overline{K}_2), 0, K_m) \text{ and } (\overline{K}_2, V(\overline{K}_2), \ell_0 + 1, K_m) & \text{ when } 0 \in A \text{ and }\abs{A \cap \sset{1,2,3}} \le 1;\\
        (\overline{K}_2, V(\overline{K}_2), 1, K_m) \text{ and } (\overline{K}_2, V(\overline{K}_2), \ell_0 + 2, K_m) & \text{ when } 0 \not\in A \text{ and }\abs{A \cap \sset{1,2,3}} = 1;\\
        (\overline{K}_2, V(\overline{K}_2), K_m) \text{ and } (\overline{K}_2, V(\overline{K}_2), \ell_0, K_m) & \text{ when } \abs{A \cap \sset{1,2,3}} \ge 2,
    \end{align*}
    and the clique extension $(D, A, K_m)$ and the path-clique extension $(D, A, \ell_0, K_m)$ respectively contain
    \begin{align*}
        (\overline{K}_2, V(\overline{K}_2), 0, K_m) \text{ and } (\overline{K}_2, V(\overline{K}_2), \ell_0 + 1, K_m) & \text{ when } A \subseteq \sset{0,2} \text{ and }A \neq \varnothing; \\
        (C_3, V_2(C_3), 0, K_m) \text{ and } (C_3, V_2(C_3), \ell_0 + 1, K_m) & \text{ when } A \in \sset{\sset{1}, \sset{3}}; \\
        (\overline{K}_2, V(\overline{K}_2), K_m) \text{ and } (\overline{K}_2, V(\overline{K}_2), \ell_0, K_m) & \text{ when } A \supseteq \sset{1,3}; \\
        (C_3, V_2(C_3), K_m) \text{ and } (C_3, V_2(C_3), \ell_0, K_m) & \text{ when } \abs{A \cap \sset{0,2}} \ge 1 \text{ and } \abs{A \cap \sset{1,3}} = 1.
    \end{align*}
    Therefore the extension families $\mathcal{X}(C, \ell, m)$ and $\mathcal{X}(D, \ell, m)$ are disjoint from $\G(\la)$.
\end{proof}

The next result shows that forbidding a star and an extension family of $F$ effectively forbids $F$ itself in every sufficiently large connected graph.

\begin{lemma} \label{lem:forbid-extensions}
    For every graph $F$ and $k, \ell, m \in \mathbb{N}^+$, there exists $N \in \mathbb{N}$ such that for every connected graph $G$ with more than $N$ vertices, if no member in $\sset{S_k} \cup \mathcal{X}(F, \ell, m)$ is a subgraph of $G$, then neither is $F$.
\end{lemma}

\begin{proof}
    With hindsight, we choose $N = d^{\ell + v}$, where $d$ is the Ramsey number $R(k, 2^{v + 1}m + v)$, and $v$ is the number of vertices of $F$. Suppose that $G$ is a connected graph with more than $N$ vertices that contains no member in $\sset{S_k} \cup \mathcal{X}(F, \ell, m)$ as a subgraph. Suppose for the sake of contradiction that the subgraph of $G$ induced on a vertex subset, say $V$, is isomorphic to $F$. Since $G$ is a connected graph with more than $d^{\ell + v}$ vertices, either the radius\footnote{Recall that the \emph{eccentricity} of a vertex $v$ in $G$ is the maximum distance between $v$ and any other vertex of $G$, and the \emph{radius} of $G$ is the minimum eccentricity of any vertex in $G$.} of $G$ is at least $\ell + v$, or the maximum degree of $G$ is at least $d$. We break the rest of the proof into two cases.

    \paragraph{Case 1:} The radius of $G$ is at least $\ell + v$. Then there exists a vertex $u$ at distance at least $\ell$ from $V$. Thus the subgraph of $G$ induced on $V$ and a shortest path from $V$ to $u$ contains the path extension $(F, A, \ell)$ as a subgraph for some nonempty $A \subseteq V$, which is a contradiction.

    \paragraph{Case 2:} The maximum degree of $G$ is at least $d$. Let $u$ be a vertex of $G$ with the maximum degree, and let $N(u)$ be the set of neighbors of $u$ in $G$. Since $S_k$ is not a subgraph of $G$, the subgraph $G[N(u)]$ cannot have an independent set of size $k$. Because $d = R(k, 2^{v + 1}m + v)$, the subgraph $G[N(u)]$ contains a clique of size $2^{v + 1}m + v$, and so there exists a clique $G[U]$ of size $2^{v + 1}m$ that is vertex-disjoint from $F$.

    We further partition the vertices in $U$ as follows. For every subset $A$ of $V$, let $U(A)$ be the set of vertices in $U$ that are adjacent to all vertices in $A$ and not adjacent to any vertex in $V \setminus A$. By the pigeonhole principle, there exists a subset $A$ of $V$ such that $\abs{U(A)} \ge 2m$. If $A$ is nonempty, then $G[V \cup U(A)]$ contains the clique extension $(F, A, K_m)$ as a subgraph, which is a contradiction.

    Hereafter we may assume that $\abs{U(\varnothing)} \ge 2m$, hence the distance between $V$ and $U(\varnothing)$ is at least $2$. Let $v_0 \dots v_{\ell_0}$ be a shortest path between $V$ and $U(\varnothing)$, where $v_0$ and $v_{\ell_0}$ are respectively at distance $1$ from $V$ and $U(\varnothing)$. Let $A_1$ be the nonempty set of vertices in $V$ that are adjacent to $v_0$, and denote by $v_{\ell_0 + 1}$ an arbitrary vertex in $U(\varnothing)$ that is adjacent to $v_{\ell_0}$. We may assume that $\ell_0 < \ell - 1$ because otherwise $G[V \cup \sset{v_0, \dots, v_{\ell_0}, v_{\ell_0 + 1}}]$ would contain the path extension $(F, A_1, \ell)$ as a subgraph, which is a contradiction. Let $A_2$ be the nonempty set of vertices in $U(\varnothing)$ that are adjacent to $v_{\ell_0}$. If $\abs{A_2} \ge m$, then $G[V \cup \sset{v_0, \dots, v_{\ell_0}} \cup A_2]$ contains the path-clique extension $(F, A_1, \ell_0, K_m)$ as a subgraph. Otherwise $\abs{U(\varnothing) \setminus A_2} > m$, and so $G[V \cup \sset{v_0, \dots, v_{\ell_0}, v_{\ell_0 + 1}} \cup (U(\varnothing) \setminus A_2)]$ contains the path-clique extension $(F, A_1, \ell_0 + 1, K_m)$.
\end{proof}

Combining \cref{lem:hoffman-extension,lem:forbid-extensions}, we obtain a finite set of forbidden subgraphs for $\G(\la)$ that forbids the claw graph and the diamond graph in every sufficiently large connected graphs. The following result, which is an immediate consequence of \cite[Theorem~4]{RW65}, gives a sufficient condition for line graphs.

\begin{theorem}[Theorem~4 of van Rooij and Wilf~\cite{RW65}] \label{thm:claw-diamond}
    Every graph that contains neither the claw graph nor the diamond graph as a subgraph is a line graph. \qed
\end{theorem}

The last ingredient for the proof of \cref{thm:main1} for $\lambda < 2$ is the following lemma attributed to Dickson, who used it to prove a result about perfect numbers in number theory.

\begin{lemma}[Lemma~A of Dickson~\cite{D13}] \label{lem:dickson}
    The partially ordered set $(\mathbb{N}^n, \le)$, in which $(a_1, \dots a_n) \le (b_1, \dots, b_n)$ if and only if $a_i \le b_i$ for every $i$, does not contain infinite antichains.
\end{lemma}

Dickson gave two proofs of \cref{lem:dickson}, one of which uses induction, while the other uses Hilbert's basis theorem. For the reader's convenience, we provide a short combinatorial proof based on the infinite Ramsey's theorem.

\begin{proof}
    Suppose that $s_1, s_2, \dots$ is an infinite sequence of distinct tuples in $\mathbb{N}^n$. For every $i < j$, color the edge $s_i s_j$ by $0$ if $s_i < s_j$, otherwise by any $k \in \sset{1, \dots, n}$ such that $s_{i,k} > s_{j,k}$. The infinite Ramsey's theorem provides an infinite subset $I \subseteq \mathbb{N}^+$ such that the edges $s_i s_j$, where $i, j \in I$ and $i\neq j$, all receive the same color $c$. Because $(\mathbb{N}, \le)$ does not contain infinite descending chains, it must be the case that $c = 0$. This implies that $\dset{s_i}{i \in I}$ is an infinite ascending chain, and in particular $\sset{s_1, s_2, \dots}$ is not an antichain.
\end{proof}

We are ready to establish the first main theorem for $\lambda < 2$.

\begin{proof}[Proof of \cref{thm:main1} for $\lambda < 2$]
    We obtain $\ell, m \in \mathbb{N}^+$ from \cref{lem:hoffman} and \cref{lem:hoffman-extension} such that the following family
    \begin{align*}
        \mathcal{F}_1 := & \sset{S_4, P_\ell} \cup \mathcal{X}(C, \ell, m) \cup \mathcal{X}(D, \ell, m) \\
        & \cup \dset{(C_n, V_2(C_n), \ell_0, K_m)}{n \in \sset{3, \dots, \ell + 1}, \ell_0 \in \sset{0, \dots, \ell - 1}} \\
        & \cup \dset{(C_n, V_2(C_n), K_m)}{n \in \sset{3, \dots, \ell + 1}} \\
        & \cup \dset{(K_m, V(K_m), \ell_0, K_m)}{\ell_0 \in \sset{0, \dots, \ell-1}}
    \end{align*}
    is disjoint from $\G(\la)$.

    \begin{claim*}
        There exists $N \in \mathbb{N}$ such that for every connected graph $G$ with more than $N$ vertices, if $G$ contains no member in $\mathcal{F}_1$ as a subgraph, then there exists a rooted tree $H \in \mathcal{T}_N$ such that $G = L(H)$, where $\mathcal{T}_N$ is the family of rooted trees such that every connected component obtained from removing the root has at most $N$ vertices.
    \end{claim*}

    \begin{claimproof}[Proof of Claim]
        We obtain $N_0$ from \cref{lem:forbid-extensions} such that for every connected graph $G$ with more than $N_0$ vertices, if no member in $\sset{S_4} \cup \mathcal{X}(C, \ell, m) \cup \mathcal{X}(D, \ell, m)$ is a subgraph of $G$, then neither is $C$ nor $D$. With hindsight, we choose $N = \max(N_0, (m+1)^{\ell+2})$. Suppose that $G$ is a connected graph with more than $N$ vertices, and suppose that no member in $\mathcal{F}_1$ is a subgraph of $G$. By our choice of $N_0$, we know that neither $C$ nor $D$ is a subgraph of $G$.

        \cref{thm:claw-diamond} implies that $G$ is the line graph of another connected graph, say $H$. Since $P_\ell$ is not a subgraph of $G$, the path $P_{\ell + 1}$ cannot be a general subgraph of $H$. In particular, the radius of $H$ is at most $\ell$. Let $d$ be the maximum degree of $H$. Suppose for a moment that $d < m + 2$. Then $H$ has at most $(m+1)^{\ell+1}$ vertices, and at most $(m+1)^{\ell+2}$ edges, and so $G$ has at most $(m+1)^{\ell+2}$ vertices, which is a contradiction.

        We may assume that $d \ge m + 2$. Let $u$ be a vertex of $H$ with degree $d$. We claim that $H$ is a tree. Suppose on the contrary that $H$ contains a cycle as a general subgraph. Since $H$ does not contain $P_{\ell + 1}$ as a general subgraph, the length of any cycle in $H$ cannot exceed $\ell + 1$. Suppose in addition that $u$ is on a cycle in $H$. Let $C_n$ be a shortest cycle containing $u$, where $n \in \sset{3, \dots, \ell + 1}$. By the choice of $C_n$, $u$ has at least $d - 2 \ge m$ neighbors outside $C_n$. Thus $H$ contains $C_n$ with $S_m$ attached at $u$ as a general subgraph, and so $G$ contains $(C_n, V_2(C_n), K_m)$ as a subgraph, which is a contradiction. Therefore $u$ is not on any cycle in $H$. Let $C_n$ be any cycle in $H$, where $n \in \sset{3, \dots, \ell + 1}$. Let $P_{\ell_0}$ be a shortest path between $u$ and $C_n$, where $\ell_0 \in \sset{1, \dots, \ell}$. Since $u$ is not on any cycle, $u$ has at least $d - 1 > m$ neighbors outside $C_n$ and $P_{\ell_0}$. Thus $H$ contains $C_n \cup P_{\ell}$ with $S_m$ attached at $u$ as a general subgraph, and so $G$ contains $(C_n, V_2(C_n), \ell_0 - 1, K_m)$ as a subgraph, which is a contradiction.

        We now view $H$ as a tree rooted at $u$. We claim that $u$ is the only vertex with degree larger than $m$. Suppose on the contrary that there is another vertex $u'$ with degree at least $m + 1$ in $H$. Let $P_{\ell_0}$ be a shortest path between $u$ and $u'$, where $\ell_0 \in \sset{1, \dots, \ell}$. Thus $H$ contains $P_{\ell_0}$ with two vertex-disjoint stars $S_m$ respectively attached to $u$ and $u'$ as a (general) subgraph, and so $G$ contains $(K_m, V(K_m), \ell_0 - 1, K_m)$ as a subgraph, which is a contradiction. Therefore all the vertices but $u$ in $H$ has degree at most $m$. After the root $u$ is removed from $H$, each connected component has radius at most $\ell$ and degree at most $m$, and so each connected component has at most $m^{\ell} \le N$ vertices.
    \end{claimproof}

    Let $N \in \mathbb{N}$ be given by the above claim, and set
    \begin{align*}
        \mathcal{F}_0 & := \dset{G \not\in \G(\la)}{G \text{ has at most }N \text{ vertices}}, \\
        \widetilde{\mathcal{F}}_2 & := \dset{G \not\in \G(\la)}{\exists H \in \mathcal{T}_N \text{ s.t.\ }G = L(H)}.
    \end{align*}
    Setting $\mathcal{F}_2$ the family of graphs that are minimal in $\widetilde{\mathcal{F}}_2$ under taking subgraphs, one can check that $\mathcal{F}_0 \cup \mathcal{F}_1 \cup \mathcal{F}_2$ is a forbidden subgraph characterization of $\G(\la)$.

    It suffices to prove that $\mathcal{F}_2$ is finite. Let $T_1, \dots, T_n$ be an enumeration of rooted trees on at most $N$ vertices. We encode $G \in \mathcal{F}_2$ by $t_G \in \mathbb{N}^n$ as follows. Let $H$ be the rooted tree in $\mathcal{T}_N$ such that $G = L(T)$. After removing the root $u$ from $H$, we view each connected component as a tree rooted at the vertex that is a child of $u$ in $H$. Set $t_G := (t_1, \dots, t_n)$, where $t_i$ is the number of occurrences of $T_i$ as a connected component in the graph obtained by removing $u$ from $H$. Because no member of $\mathcal{F}_2$ is a subgraph of any other, one can deduce that $\dset{t_G}{G \in \mathcal{F}_2}$ is an antichain in $(\mathbb{N}^n, \le)$, and so $\mathcal{F}_2$ is finite by \cref{lem:dickson}.
\end{proof}

\subsection{Proof of \cref{thm:main1} for $\lambda \in [2, \lambda^*)$} \label{sec:main1-2}

We prove a stronger result from which the first main theorem for $\lambda \in [2, \lambda^*)$ and its corollary for $\lambda \in (2,\lambda^*)$ follow.

\begin{theorem} \label{thm:main1-2}
    For every $\lambda \in [2, \lambda^*)$, the number of connected graphs in $\G(\la) \setminus \mathcal{G}(2)$ is finite.
\end{theorem}

\begin{proof}[Proof of \cref{thm:main1} for $\lambda \in [2, \lambda^*)$ assuming \cref{thm:main1-2}]
    It is known that $\mathcal{G}(2)$ has a finite forbidden subgraph characterization --- in fact, there are $1812$ minimal forbidden subgraphs for $\mathcal{G}(2)$ (see \cite{BN92}). In view of \cref{thm:main1-2}, it suffices to show that for every minimal forbidden subgraph $G$ for $\G(\la)$, $G$ is a minimal forbidden subgraph for $\mathcal{G}(2)$, or $G$ after removing some vertex is a connected graph in $\G(\la) \setminus \mathcal{G}(2)$.

    Indeed, let $G$ be a minimal forbidden subgraph for $\G(\la)$. Because $\lambda_1(G) < -\lambda \le -2$, $G$ contains a minimal forbidden subgraph for $\mathcal{G}(2)$ as a subgraph, say $F$. We may assume that $F$ is a proper subgraph of $G$ for otherwise we are done already. Consider the graph $H$ obtained after removing $v$ from $G$, where $v$ is a vertex of $G$ that is furthest from $F$. One can then check that $H$ is connected. Finally $H \in \G(\la)$ because of the minimality of $G$, and $H \not\in \mathcal{G}(2)$ because $F$ is a subgraph of $H$.
\end{proof}

\begin{proof}[Proof of \cref{cor:main1} for $\lambda \in (2,\lambda^*)$]
    It follows immediately from \cref{thm:main1-2} that $-\lambda$ is not a limit point of the set of smallest eigenvalues of graphs for $\lambda \in (2,\lambda^*)$
\end{proof}

The proof of \cref{thm:main1-2} centers around the notion of \emph{generalized line graphs}. Although we do not need the definition of generalized line graphs, we state it nevertheless for concreteness.

\begin{definition}[Cocktail party graphs and generalized line graphs] \label{def:gen-line-graph}
    The \emph{cocktail party graph} $\overline{a P_2}$ is obtained from the complete graph on $2a$ vertices by deleting a perfect matching. Given a graph $G$ with vertices $v_1, \dots, v_n$, and $a_1, \dots a_n \in \mathbb{N}$, the \emph{generalized line graph} $L(G; a_1, \dots, a_n)$ is obtained from the line graph $L(G)$ of $G$ by adjoining $n$ vertex-disjoint cocktail party graphs $\overline{a_1P_2}, \dots, \overline{a_nP_2}$ where every vertex of the $i$th cocktail party graph $\overline{a_iP_2}$ is adjacent to every vertex of $L(G)$ that contains $v_i$. See \cref{fig:generalized-line-graph} for an example.
\end{definition}

\begin{figure}[t]
    \centering
    \begin{tikzpicture}[very thick, scale=0.5, baseline=(v.base)]
        \coordinate (v) at (0,0);
        \draw (-2,0) -- (0,-1) node[vertex]{} -- (2,0);
        \draw (-2,0) node[vertex]{} -- (2,0) node[vertex]{} -- (0,1) node[vertex]{} -- cycle;
        \draw (-2,0) node[left]{$v_1$};
        \draw (0,1) node[above]{$v_2$};
        \draw (0,-1.1) node[below]{$v_3$};
        \draw (2,0) node[right]{$v_4$};
    \end{tikzpicture}\qquad
    \begin{tikzpicture}[very thick, scale=0.5, baseline=(v.base)]
        \coordinate (v) at (0,0);
        \draw[fill=litegray, draw=none] (0,0) -- (-5.5,0.2) -- (-5.5,-0.2) -- cycle;
        \draw[fill=litegray, draw=none] (-3,1.5) -- (-5.5,0.25) -- (-5.5,-0.3) -- cycle;
        \draw[fill=litegray, draw=none] (-3,-1.5) -- (-5.5,0.3) -- (-5.5,-0.25) -- cycle;
        \draw[fill=litegray, draw=none] (-3,1.5) -- (0,3.3) -- (0,2.8) -- cycle;
        \draw[fill=litegray, draw=none] (3,1.5) -- (0,3.3) -- (0,2.8) -- cycle;
        \draw[fill=litegray, draw=none] (0,0) -- (6,0.22) -- (6,-0.22) -- cycle;
        \draw[fill=litegray, draw=none] (3,1.5) -- (6,0.27) -- (6,-0.32) -- cycle;
        \draw[fill=litegray, draw=none] (3,-1.5) -- (6,0.32) -- (6,-0.27) -- cycle;
        \draw (-3,1.5) node[vertex]{} -- (3,1.5) -- (3,-1.5) node[vertex]{} -- (-3,-1.5) -- (-3,1.5) -- (3,-1.5);
        \draw (-3,-1.5) node[vertex]{} -- (0,0) node[vertex]{} -- (3,1.5)node[vertex]{};
        \node[below] at (0,-0.3) {$v_1v_4$};
        \node[above] at (-3.5,1.5) {$v_1v_2$};
        \node[above] at (3.5,1.5) {$v_2v_4$};
        \node[below] at (-3.5,-1.6) {$v_1v_3$};
        \node[below] at (3.5,-1.6) {$v_3v_4$};
        \draw[fill=litegray, draw=none] (-5.5, 0) circle (1.1);
        \draw (-5, -0.5) node[vertex]{} -- (-5,0.5) node[vertex]{}-- (-6,0.5) node[vertex]{} -- (-6,-0.5) node[vertex]{} -- cycle;
        \draw[fill=litegray, draw=none] (0, 3) circle (0.9);
        \draw (-0.5,3) node[vertex]{};
        \draw (0.5,3) node[vertex]{};
        \draw[fill=litegray, draw=none] (6, 0) circle (1.4);
        \draw (7,0) -- ++(150:{sqrt(3)}) -- ++(0,{-sqrt(3)}) -- cycle;
        \draw (5,0) -- ++(30:{sqrt(3)}) -- ++(0,{-sqrt(3)}) -- cycle;
        \draw (7,0) node[vertex]{} -- ++(120:1) node[vertex]{} -- ++(-1,0) node[vertex]{} -- ++(240:1) node[vertex]{} -- ++(-60:1) node[vertex]{} -- ++(1,0) node[vertex]{} -- cycle;
    \end{tikzpicture}
    \caption{A graph $G$ and a schematic drawing of its generalized line graph $L(G; 2, 1, 0, 3)$.} \label{fig:generalized-line-graph}
\end{figure}

Just like line graphs, all the generalized line graphs are in $\mathcal{G}(2)$, and they have a finite forbidden subgraph characterization.

\begin{theorem}[Theorem~2.1 of Hoffman~\cite{H77b}] \label{thm:generalized-line-smallest-ev}
    The smallest eigenvalue of a generalized line graph is at least $-2$. \qed
\end{theorem}

\begin{theorem}[Cvetkovi\'{c} et al.~\cite{CDS80,CDS81}, and Rao et al.~\cite{RSV81}] \label{thm:forb-char-generalized-line}
    The minimal forbidden subgraphs for the family $\mathcal{D}_\infty$ of generalized line graphs are listed in \cref{fig:minimal-forb-1}. \qed
\end{theorem}

\begin{figure}
    \centering
    \begin{tikzpicture}[very thick, baseline=(v.base)]
    \coordinate (v) at (0,.1);
    \coordinate (v0) at (0,0);
    \path (v0) + (90:1) coordinate (v1);
    \path (v0) + (162:1) coordinate (v2);
    \path (v0) + (234:1) coordinate (v3);
    \path (v0) + (306:1) coordinate (v4);
    \path (v0) + (378:1) coordinate (v5);
    \draw (v4) -- (v5);
    \draw (v3) -- (v2);
    \draw (v1) -- (v0);
    \draw (v1) -- (v2);
    \draw (v1) -- (v5);
    \draw (v0) -- (v2);
    \draw (v0) -- (v5);
    \node[circ] at (v4) {0};
    \node[circ] at (v3) {1};
    \node[circ] at (v1) {2};
    \node[circ] at (v0) {3};
    \node[circ] at (v2) {4};
    \node[circ] at (v5) {5};
    \node at (0,-1.4) {$G_{1}$};
    \end{tikzpicture}
    \begin{tikzpicture}[very thick, baseline=(v.base)]
    \coordinate (v) at (0,.1);
    \coordinate (v0) at (0,0);
    \path (v0) + (90:1) coordinate (v1);
    \path (v0) + (162:1) coordinate (v2);
    \path (v0) + (234:1) coordinate (v3);
    \path (v0) + (306:1) coordinate (v4);
    \path (v0) + (378:1) coordinate (v5);
    \draw (v4) -- (v3);
    \draw (v4) -- (v5);
    \draw (v3) -- (v2);
    \draw (v1) -- (v0);
    \draw (v1) -- (v2);
    \draw (v1) -- (v5);
    \draw (v0) -- (v2);
    \draw (v0) -- (v5);
    \node[circ] at (v4) {0};
    \node[circ] at (v3) {1};
    \node[circ] at (v1) {2};
    \node[circ] at (v0) {3};
    \node[circ] at (v2) {4};
    \node[circ] at (v5) {5};
    \node at (0,-1.4) {$G_{2}$};
    \end{tikzpicture}
    \begin{tikzpicture}[very thick, baseline=(v0.base)]
    \coordinate (v0) at (0,0);
    \path (v0) + (90:1) coordinate (v1);
    \path (v0) + (180:1) coordinate (v2);
    \path (v0) + (270:1) coordinate (v3);
    \path (v0) + (360:1) coordinate (v4);
    \draw (v1) -- (v2);
    \draw (v1) -- (v4);
    \draw (v3) -- (v0);
    \draw (v3) -- (v2);
    \draw (v3) -- (v4);
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    \end{tikzpicture}
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    \draw (v1) -- (v3);
    \node[circ] at (v4) {0};
    \node[circ] at (v5) {1};
    \node[circ] at (v0) {2};
    \node[circ] at (v2) {3};
    \node[circ] at (v1) {4};
    \node[circ] at (v3) {5};
    \node at (0,-1.5) {$G_{25}$};
    \end{tikzpicture}
    \begin{tikzpicture}[very thick, baseline=(v.base)]
    \coordinate (v) at (0,0);
    \coordinate (v0) at (0,1);
    \coordinate (v1) at (-1,0);
    \coordinate (v2) at (0,-1);
    \coordinate (v3) at (1,0);
    \coordinate (v4) at (0,.4);
    \coordinate (v5) at (0,-.4);
    \draw (v2) -- (v5);
    \draw (v0) -- (v4);
    \draw (v1) -- (v3);
    \draw (v1) -- (v4);
    \draw (v3) -- (v4);
    \draw (v5) -- (v4);
    \node[circ] at (v2) {0};
    \node[circ] at (v0) {1};
    \node[circ] at (v1) {2};
    \node[circ] at (v3) {3};
    \node[circ] at (v5) {4};
    \node[circ] at (v4) {5};
    \node at (0,-1.5) {$G_{26}$};
    \end{tikzpicture}
    \begin{tikzpicture}[very thick, baseline=(v.base)]
    \coordinate (v) at (0,.1);
    \coordinate (v0) at (0,0);
    \path (v0) + (90:1) coordinate (v1);
    \path (v0) + (162:1) coordinate (v2);
    \path (v0) + (234:1) coordinate (v3);
    \path (v0) + (306:1) coordinate (v4);
    \path (v0) + (378:1) coordinate (v5);
    \draw (v4) -- (v0);
    \draw (v3) -- (v2);
    \draw (v3) -- (v0);
    \draw (v1) -- (v5);
    \draw (v1) -- (v2);
    \draw (v1) -- (v0);
    \draw (v5) -- (v2);
    \draw (v5) -- (v0);
    \draw (v2) -- (v0);
    \node[circ] at (v4) {0};
    \node[circ] at (v3) {1};
    \node[circ] at (v1) {2};
    \node[circ] at (v5) {3};
    \node[circ] at (v2) {4};
    \node[circ] at (v0) {5};
    \node at (0,-1.4) {$G_{27}$};
    \end{tikzpicture}
    \begin{tikzpicture}[very thick, baseline=(v.base)]
    \coordinate (v) at (0,.1);
    \coordinate (v0) at (0,0);
    \path (v0) + (90:1) coordinate (v1);
    \path (v0) + (162:1) coordinate (v2);
    \path (v0) + (234:1) coordinate (v3);
    \path (v0) + (306:1) coordinate (v4);
    \path (v0) + (378:1) coordinate (v5);
    \draw (v4) -- (v5);
    \draw (v4) -- (v0);
    \draw (v3) -- (v2);
    \draw (v3) -- (v0);
    \draw (v1) -- (v2);
    \draw (v1) -- (v5);
    \draw (v1) -- (v0);
    \draw (v2) -- (v5);
    \draw (v2) -- (v0);
    \draw (v5) -- (v0);
    \node[circ] at (v4) {0};
    \node[circ] at (v3) {1};
    \node[circ] at (v1) {2};
    \node[circ] at (v2) {3};
    \node[circ] at (v5) {4};
    \node[circ] at (v0) {5};
    \node at (0,-1.4) {$G_{28}$};
    \end{tikzpicture}
    \begin{tikzpicture}[very thick, baseline=(v.base)]
    \coordinate (v) at (0,.1);
    \coordinate (v0) at (0,0);
    \path (v0) + (90:1) coordinate (v1);
    \path (v0) + (162:1) coordinate (v2);
    \path (v0) + (234:1) coordinate (v3);
    \path (v0) + (306:1) coordinate (v4);
    \path (v0) + (378:1) coordinate (v5);
    \draw (v4) -- (v5);
    \draw (v4) -- (v0);
    \draw (v3) -- (v2);
    \draw (v3) -- (v0);
    \draw (v1) -- (v2);
    \draw (v1) -- (v5);
    \draw (v1) -- (v0);
    \draw (v2) -- (v0);
    \draw (v5) -- (v0);
    \node[circ] at (v4) {0};
    \node[circ] at (v3) {1};
    \node[circ] at (v1) {2};
    \node[circ] at (v2) {3};
    \node[circ] at (v5) {4};
    \node[circ] at (v0) {5};
    \node at (0,-1.4) {$G_{29}$};
    \end{tikzpicture}
    \begin{tikzpicture}[very thick, baseline=(v.base)]
    \coordinate (v) at (0,.1);
    \coordinate (v0) at (0,0);
    \path (v0) + (90:1) coordinate (v1);
    \path (v0) + (162:1) coordinate (v2);
    \path (v0) + (234:1) coordinate (v3);
    \path (v0) + (306:1) coordinate (v4);
    \path (v0) + (378:1) coordinate (v5);
    \draw (v1) -- (v0);
    \draw (v5) -- (v4);
    \draw (v5) -- (v0);
    \draw (v2) -- (v3);
    \draw (v2) -- (v0);
    \draw (v3) -- (v0);
    \draw (v4) -- (v0);
    \node[circ] at (v1) {0};
    \node[circ] at (v5) {1};
    \node[circ] at (v2) {2};
    \node[circ] at (v3) {3};
    \node[circ] at (v4) {4};
    \node[circ] at (v0) {5};
    \node at (0,-1.4) {$G_{30}$};
    \end{tikzpicture}
    \begin{tikzpicture}[very thick, baseline=(v.base)]
    \coordinate (v) at (0,.1);
    \coordinate (v0) at (0,0);
    \path (v0) + (90:1) coordinate (v1);
    \path (v0) + (162:1) coordinate (v2);
    \path (v0) + (234:1) coordinate (v3);
    \path (v0) + (306:1) coordinate (v4);
    \path (v0) + (378:1) coordinate (v5);
    \draw (v1) -- (v0);
    \draw (v5) -- (v4);
    \draw (v5) -- (v0);
    \draw (v2) -- (v3);
    \draw (v2) -- (v0);
    \draw (v3) -- (v4);
    \draw (v3) -- (v0);
    \draw (v4) -- (v0);
    \node[circ] at (v1) {0};
    \node[circ] at (v5) {1};
    \node[circ] at (v2) {2};
    \node[circ] at (v3) {3};
    \node[circ] at (v4) {4};
    \node[circ] at (v0) {5};
    \node at (0,-1.4) {$G_{31}$};
    \end{tikzpicture}
    \caption{Minimal forbidden subgraphs for $\mathcal{D}_\infty$.} \label{fig:minimal-forb-1}
\end{figure}

The key observation for the proof of \cref{thm:main1-2} is that an extension family of $F$ is disjoint from $\G(\la)$ for every minimal forbidden subgraph $F$ for the family $\mathcal{D}_\infty$. We first deal with path extensions and clique extensions.

\begin{lemma} \label{lem:computation-1}
    For every minimal forbidden subgraph $F$ in \cref{fig:minimal-forb-1} for the family $\mathcal{D}_\infty$ of generalized line graphs, and every nonempty vertex subset $A$ of $F$, the path extensions and the clique extensions of $F$ satisfy
    \[
        \lim_{\ell\to\infty} \lambda_1(F, A, \ell) \le -\lambda^*
        \quad \text{and} \quad
        \lim_{m\to\infty} \lambda_1(F, A, K_m) \le -\lambda^*.
    \]
\end{lemma}

Notice that the path extensions $(G_{23}, \sset{1}, \ell)$ and $(G_{23}, \sset{2}, \ell)$ are isomorphic to $E_{2, \ell+3}$ (see \cref{fig:e2n}), and so equality holds in the first inequality of \cref{lem:computation-1} for $F = G_{23}$ and $A = \sset{1}$ or $\sset{2}$ (see \cref{lem:e2n}). In \cref{sec:numerics}, we prove, under computer assistance, that the inequalities in \cref{lem:computation-1} are strict for the remaining cases.

The next result takes care of path-clique extensions.

\begin{lemma} \label{lem:path-clique-extension}
    Suppose that $A$ is a nonempty vertex subset of a graph $F$ and $\lambda \ge 2$. If the path extensions of $F$ satisfy
    \[
        \lim_{\ell\to\infty} \lambda_1(F, A, \ell) < -\lambda,
    \]
    then there exists $m \in \mathbb{N}^+$ such that the path-clique extensions of $F$ satisfy
    \[
        \lambda_1(F, A, \ell, K_m) < -\lambda \text{ for every }\ell \in \mathbb{N}.
    \]
\end{lemma}

\begin{proof}
    Pick $\ell_0 \in \mathbb{N}$ such that $\lambda_1(F, A, \ell) < -\lambda$ for every $\ell \ge \ell_0$. Clearly $\lambda_1(F, A, \ell, K_m) < -\lambda$ for every $\ell \ge \ell_0$ and every $m \in \mathbb{N}^+$. It suffices to show that for every $\ell \in \sset{0, \dots, \ell_0 - 1}$ there exists $m \in \mathbb{N}^+$ such that $\lambda_1(F, A, \ell, K_m) < -\lambda$.

    Let $v_0 \dots v_{\ell_0}$ be the path added to $F$ to obtain $(F, A, \ell_0)$, where the vertex $v_0$ is connected to every vertex in $A$. Set $\lambda_0 := -\lambda_1(F, A, \ell_0)$, and let $\bm{x} \colon V(F) \cup \sset{v_0, \dots, v_{\ell_0}} \to \mathbb{R}$ be an eigenvector of $(F, A, \ell_0)$ associated with $-\lambda_0$.

    Now fix $\ell \in \mathbb{N}$ with $\ell < \ell_0$, and let $m \in \mathbb{N}^+$ be determined later. Set $V_\ell := V(F) \cup \sset{v_0, \dots, v_\ell}$, and identify the vertex set of $(F, A, \ell, K_m)$ with $V_\ell \cup V(K_m)$. Define $\tilde{\bx}\colon V_\ell \cup V(K_m) \to \mathbb{R}$ by
    \[
        \tilde{x}_v = \begin{cases}
            x_v & \text{if }v \in V_\ell; \\
            x_{\ell+1}/m & \text{if }v \in V(K_m).
        \end{cases}
    \]
    We claim that
    \[
        \sum_{v \in V_\ell}x_v^2 > 0,
    \]
    which in particular implies that $\tilde{\bx}$ is a nonzero vector. Indeed, assume for the sake of contradiction that $x_v = 0$ for $v \in V_\ell$. Using $-\lambda_0x_i = \sum_{u\sim i}x_u$ for $i \in \sset{\ell, \dots, \ell_0-1}$, where the sum is taken over all vertices $u$ that are adjacent to the vertex $i$ in $(F, A, \ell_0)$, one can prove inductively that $x_i = 0$ for $i \in \sset{\ell + 1, \dots, \ell_0}$, which contradicts with the assumption that $\bm{x}$ is a nonzero vector.

    The Rayleigh principle says that $\lambda_1(F, A, \ell, K_m)$ is at most the Rayleigh quotient
    \[
        \frac{\tilde{\bx}^\intercal A_{(F, A, \ell, K_m)} \tilde{\bx}}{\tilde{\bx}^\intercal \tilde{\bx}} = \frac{\sum_{v \in V_\ell} x_v \sum_{u \sim v} x_u + x_\ell x_{\ell+1} + m(m-1)(x_{\ell+1}/m)^2}{\sum_{v \in V_\ell}x_v^2 + m(x_{\ell + 1}/m)^2}.
    \]
    Since $\bm{x}$ is an eigenvector of $(F, A, \ell_0)$ associated with $-\lambda_0$, we obtain that $\sum_{u \sim v} x_u = -\lambda_0 x_v$. Thus the Rayleigh quotient can be simplified to
    \[
        \frac{-\lambda_0 \sum_{v \in V_\ell} x_v^2 + x_\ell x_{\ell+1} + m(m-1)(x_{\ell+1}/m)^2}{\sum_{v \in V_\ell}x_v^2 + m(x_{\ell + 1}/m)^2},
    \]
    which, as $m \to \infty$, approaches
    \[
        -\lambda_0 + \frac{(x_\ell + x_{\ell+1})x_{\ell+1}}{\sum_{v \in V_\ell}x_v^2}.
    \]

    Recall that $\lambda_0 = -\lambda_1(F, A, \ell_0) > \lambda \ge 2$. It suffices to show that $(x_\ell + x_{\ell+1})x_{\ell+1} \le 0$. In fact, we prove inductively that $(x_i + x_{i+1})x_{i+1} \le 0$ for $i \in \sset{\ell, \dots, \ell_0 - 1}$. The base case where $i = \ell_0 - 1$ follows immediately from $-\lambda_0x_{\ell_0} = x_{\ell_0-1}$ and $\lambda_0 > 2$. For the inductive step, using $-\lambda_0x_{i+1} = x_i + x_{i+2}$ and $\lambda_0 > 2$, we obtain
    \begin{multline*}
        (x_i + x_{i+1})x_{i+1} = (-\lambda_0x_{i+1} - x_{i+2} + x_{i+1})x_{i+1} \le (-x_{i+1}-x_{i+2})x_{i+1} \\
        = -(x_{i+1}+x_{i+2})^2 + (x_{i+1}+x_{i+2})x_{i+2} \le (x_{i+1}+x_{i+2})x_{i+2},
    \end{multline*}
    which is nonpositive by the inductive hypothesis.
\end{proof}

We are ready to prove \cref{thm:main1-2}.

\begin{proof}[Proof of \cref{thm:main1-2}]
    Suppose that $\lambda \in [2, \lambda^*)$. Denote $\mathcal{F}$ the set of minimal forbidden subgraphs for the family $\mathcal{D}_\infty$. Combining \cref{lem:computation-1,lem:path-clique-extension}, we choose $\ell, m \in \mathbb{N}^+$ such that for every $F \in \mathcal{F}$, the extension family $\mathcal{X}(F, \ell, m)$ is disjoint from $\G(\la)$. From \cref{lem:hoffman}(\ref{lem:s}), we know that $S_5 \not\in \G(\la)$. In particular, no graph in $\G(\la)$ contains any member in the following family
    \begin{equation} \label{eqn:main1-2}
        \sset{S_5} \cup \bigcup_{F \in \mathcal{F}} \mathcal{X}(F, \ell, m)
    \end{equation}
    as a subgraph. Using \cref{lem:forbid-extensions}, we obtain $N \in \mathbb{N}$ such that for every connected graph $G$ with more than $N$ vertices, if no member in \eqref{eqn:main1-2} is a subgraph of $G$, then neither is any $F \in \mathcal{F}$, and so $G$ is a generalized line graph and is in $\mathcal{G}(2)$ by \cref{thm:forb-char-generalized-line,thm:generalized-line-smallest-ev}. This implies that every connected graph in $\G(\la) \setminus \mathcal{G}(2)$ has at most $N$ vertices.
\end{proof}

We end this subsection with a remark on an alternative proof of \cref{thm:main1-2}. Instead of working with the family $\mathcal{D}_\infty$, one should be able to establish \cref{lem:computation-1} for every minimal forbidden subgraph for $\mathcal{G}(2)$, and then prove \cref{thm:main1-2} similarly. This alternative approach is more direct as it does not rely on the generalized line graphs. However the drawback is obvious --- there are $1812$ minimal forbidden subgraphs for $\mathcal{G}(2)$. To the best of our knowledge, there is no easily accessible database for these $1812$ graphs; for example, a complete list of these graphs was printed to microfiche in \cite{BN92}, and a list of only $92$ of these graphs up to $8$ vertices was available in \cite[Fig.~2.4 and Table~A1.2]{CRS04}. One certainly can implement the reasonably fast algorithm in \cite{BN92} to enumerate the minimal forbidden subgraphs for $\mathcal{G}(2)$. However as we try to keep the computer-assisted part of the proof to a minimum, we work with the $31$ minimal forbidden subgraphs for $\mathcal{D}_\infty$.

In fact, the family $\mathcal{G}(2)$ and its subfamily $\mathcal{D}_\infty$ are interchangeable when both families are restricted to sufficiently large connected graphs because of a characterization of $\mathcal{G}(2)$ due to Cameron, Goethals, Seidel, and Shult. To state their characterization, we introduce the following definitions.

\begin{definition}[Representaiton of graphs and root systems $D_n$ and $E_8$] \label{def:dn-e8}
    Given a graph $G$ and $V \subseteq \mathbb{R}^n$, we say that $G$ is \emph{represented} by $V$ if the Gram matrix of $V$ equals $A_G + 2I$, where $A_G$ is the adjacency matrix of $G$. The \emph{root systems} $D_n$ and $E_8$ are defined by the standard basis $e_1, \dots e_n$ of $\mathbb{R}^n$ as follows
    \[
        D_n := \dset{\varepsilon_1 \bm{e}_i + \varepsilon_2 \bm{e}_j}{\varepsilon_1, \varepsilon_2 = \pm 1, 1 \le i < j \le n}, \quad E_8 := D_4 \cup \dset{\frac12\sum_{i=1}^8 \varepsilon_i \bm{e}_i}{\varepsilon_i = \pm 1, \prod_{i=1}^8 \varepsilon_i = 1}.
    \]
\end{definition}

\begin{theorem}[Theorems~4.2, 4.3 and 4.10 of Cameron et al.~\cite{CGSS76}] \label{thm:cgss}
    For every connected graph $G$, the smallest eigenvalue of $G$ is at least $-2$ if and only if $G$ is represented by a subset of $D_n$ or $E_8$. Moreover
    \begin{enumerate}[nosep, label=(\alph*)]
        \item a graph (not necessarily connected) is represented by a subset of $D_n$ if and only if it is a generalized line graph; and
        \item a graph represented by a subset of $E_8$ has at most $36$ vertices, and its maximum degree is at most $28$. \qed
    \end{enumerate}
\end{theorem}

\cref{thm:cgss} particularly implies that every connected graph in $\mathcal{G}(2)$ with more than $36$ vertices is a generalized line graph.

\subsection{Proof of \cref{thm:main1} for $\lambda \ge \lambda^*$}

Suppose that $\sset{F_1, \dots, F_n}$ is a finite forbidden subgraph characterization for $\G(\la)$. Because every graph that is not in $\G(\la)$ contains $F_i$ as a subgraph for some $i \in \sset{1, \dots, n}$, no graph has its smallest eigenvalue in the open interval $(\max\dset{\lambda_1(F_i)}{i \in \sset{1, \dots, n}},-\lambda)$. Recall that $\Lambda_1$ consists of $\lambda \in \mathbb{R}$ such that $-\lambda$ is the smallest eigenvalue of some graph. The contrapositive of the above observation says the following.

\begin{proposition} \label{lem:limit-points}
    Let $\lim_+ \Lambda_1 := \dset{\lambda \in \mathbb{R}}{(\lambda, \lambda + \varepsilon) \cap \Lambda_1 \neq \varnothing \text{ for every }\varepsilon > 0}$ be the set of right-sided limit points of $\Lambda_1$. The family $\G(\la)$ does not have a finite forbidden subgraph characterization for any $\lambda \in \lim_+ \Lambda_1$. \qed
\end{proposition}

In fact, we prove that $\Lambda_1$ is dense in $(\lambda^*, \infty)$, from which the first main theorem and its corollary for $\lambda \ge \lambda^*$ follow.

\begin{theorem} \label{thm:main1-3}
    For every $\lambda > \lambda^*$, there exist graphs $G_1, G_2, \dots$ such that $\lim \lambda_1(G_n) = -\lambda$.
\end{theorem}

\begin{proof}[Proof of \cref{thm:main1} for $\lambda \ge \lambda^*$]
    \cref{thm:main1-3} implies that $\lim_+\Lambda \supseteq [\lambda^*, \infty)$, which implies through \cref{lem:limit-points} that $\G(\la)$ has no finite forbidden subgraph characterization for any $\lambda \ge \lambda^*$.
\end{proof}

\begin{proof}[Proof of \cref{cor:main1} for $\lambda \ge \lambda^*$]
    It follows immediately from \cref{thm:main1-3} that $-\lambda$ is a limit point of the set of smallest eigenvalues of graphs for $\lambda \ge \lambda^*$.
\end{proof}

A large chunk of \cref{thm:main1-3} is essentially established by Shearer~\cite{S89}, who proved that the set of spectral radii of all graphs is dense in $[\lambda', \infty)$, where $\lambda' = \sqrt{2+\sqrt{5}}$. As was pointed out in \cite{D89}, Shearer actually proved that the set of spectral radii of all caterpillar trees\footnote{A caterpillar tree is a tree in which all the vertices are within distance 1 of a central path.} is dense in $[\lambda', \infty)$ already. Since a caterpillar tree is bipartite, we rephrase Shearer's result in terms of smallest eigenvalues.

\begin{theorem}[Shearer~\cite{S89}; cf.\ Theorem 3 of Doob~\cite{D89}] \label{thm:shearer}
    For every $\lambda \ge \lambda'$, there exist caterpillar trees $G_1, G_2, \dots$ such that $\lim \lambda_1(G_n) = -\lambda$.
\end{theorem}

To fill the gap between $\lambda^* \approx 2.01980$ and $\lambda' \approx 2.05817$, we use the following graphs.

\begin{definition}
    Given a sequence $(a_1, \dots, a_n)$ of natural numbers, a \emph{rowing graph} $R(a_1, \dots, a_n)$ is obtained from the path $v_{-2}v_{-1}v_{0}v_{1}\dots v_{n}$ (called the \emph{central path}) by attaching a vertex (called the \emph{coxswain}) to $v_0$, and attaching a clique of size $a_i$ to both $v_{i-1}$ and $v_i$ for every $i \in \sset{1, \dots, n}$. (See \cref{fig:rowing-graph} for an example of a rowing graph.)
\end{definition}

\begin{figure}[t]
    \centering
    \begin{tikzpicture}[very thick, scale=0.6]
        \draw (0,3);
        \draw (-1,2) node[vertex]{} -- (0,0);
        \draw (0.75,2) -- (1.25,2);
        \draw (0,0) -- (0.75,2) node[vertex]{} -- (2,0);
        \draw (0,0) -- (1.25,2) node[vertex]{} -- (2,0);

        \draw (4.75,2) -- (5.25,2);
        \draw (4,0) -- (4.75,2) node[vertex]{} -- (6,0);
        \draw (4,0) -- (5.25,2) node[vertex]{} -- (6,0);
        \draw (6,0) -- (7,2) node[vertex]{} -- (8,0);

        \draw[fill=litegray, draw=none] (10,0) -- (11,1.5) -- (12, 0) -- (11,2.7) -- cycle;

        \draw[fill=litegray, draw=none] (11, 2) circle (0.75);
        \draw[fill=black, draw=none] (11, 2) circle (0.3);

        \draw (11,1.608) node[vertex]{} -- ++(22.5:0.3) node[vertex]{} -- ++(67.5:0.3) node[vertex]{} -- ++(112.5:0.3) node[vertex]{} -- ++(157.5:0.3) node[vertex]{} -- ++(202.5:0.3) node[vertex]{} -- ++(247.5:0.3) node[vertex]{} -- ++(292.5:0.3) node[vertex]{} -- ++(337.5:0.3) node[vertex]{};

        \draw[fill=litegray, draw=none] (12,0) -- (13,1.6) -- (14,0) -- (13,2.6) -- cycle;

        \draw[fill=litegray, draw=none] (13, 2) circle (0.6);

        \draw (13,2.25) node[vertex]{} -- ++(240:0.433) node[vertex]{} -- ++(0:0.433) node[vertex]{} -- cycle;

        \draw (14,0) -- (15,2) node[vertex]{} -- (16,0);

        \draw (-4,0) node[vertex]{} -- (-2,0) node[vertex]{} -- (0,0) node[vertex]{} -- (2,0) node[vertex]{} -- (4,0) node[vertex]{} -- (6,0) node[vertex]{} -- (8,0) node[vertex]{} -- (10,0) node[vertex]{} -- (12,0) node[vertex]{} -- (14,0) node[vertex]{} -- (16,0) node[vertex]{};
        \draw (-4,0) node[below]{$v_{-2}$};
        \draw (-2,0) node[below]{$v_{-1}$};
        \draw (0,0) node[below]{$v_{0}$};
        \draw (2,0) node[below]{$v_1$};
        \draw (4,0) node[below]{$v_2$};
        \draw (6,0) node[below]{$v_3$};
        \draw (8,0) node[below]{$v_4$};
        \draw (10,0) node[below]{$v_5$};
        \draw (12,0) node[below]{$v_6$};
        \draw (14,0) node[below]{$v_7$};
        \draw (16,0) node[below]{$v_8$};
    \end{tikzpicture}
    \caption{A schematic drawing of the rowing graph $R(2,0,2,1,0,8,3,1)$.} \label{fig:rowing-graph}
\end{figure}

We adopt the shorthand
\[
    (a_1, \dots, a_{k-1}, 0^{(\ell)}, a_k, \dots, a_n) := (a_1, \dots, a_{k-1}, \underbrace{0, \dots, 0}_{\ell}, a_k, \dots, a_n).
\]

\begin{lemma} \label{lem:rowing}
    Rowing graphs satisfy the following properties.
    \begin{enumerate}[nosep, label=(\alph*)]
        \item \label{lem:rowing-0} The smallest eigenvalue of a rowing graph is at least $-1-\sqrt{2}$.
        \item \label{lem:rowing-a} There exists $a_1 \in \mathbb{N}$ such that $\lambda_1(R(a_1)) < -\lambda' \approx -2.05817$.
        \item \label{lem:rowing-b} For every $\varepsilon > 0$ there exists $\ell \in \mathbb{N}^+$ such that
            \begin{multline*}
                \lambda_1(R(a_1, \dots, a_{n_1}, 0^{(\ell)}, b_{1}, \dots, b_{n_2})) > \lambda_1(R(a_1, \dots, a_{n_1}, 0^{(\ell)})) - \varepsilon\\
                \text{ for every }n_1, n_2 \in \mathbb{N}, (a_1, \dots, a_{n_1}) \in \mathbb{N}^{n_1} \text{ and } (b_1, \dots, b_{n_2}) \in \mathbb{N}^{n_2}.
            \end{multline*}
        \item \label{lem:rowing-c} For every $n, \ell \in \mathbb{N}^+$ and $(a_1, \dots, a_n) \in \mathbb{N}^n$ with $a_n > 0$, if $\lambda_1(R(a_1, \dots, a_n, 0^{(\ell)})) \le -2$, then for every $\varepsilon > 0$ there exists $a_{n+1} \in \mathbb{N}^+$ such that
            \[
                \lambda_1(R(a_1, \dots, a_{n-1}, a_n-1, a_{n+1}, 0^{(\ell - 1)})) < \lambda_1(R(a_1, \dots, a_n, 0^{(\ell)})) + \varepsilon.
            \]
        \item \label{lem:rowing-d} For every $\varepsilon > 0$, there exists $m \in \mathbb{N}^+$ such that for every $n \ge m$ and every $(a_1, \dots, a_n) \in \mathbb{N}^n$ there exists $k \in \sset{1, \dots, m}$ such that
            \[
                \lambda_1(R(a_1, \dots, a_{k-1}, 0, a_k, \dots, a_n)) < \frac{\lambda_1(R(a_1, \dots, a_n)) + \varepsilon}{1 + \varepsilon / 2}.
            \]
    \end{enumerate}
\end{lemma}

\begin{proof}[Proof of \ref{lem:rowing-0}]
    Set $\bm{a} = (a_1, \dots, a_n)$ and $\lambda = \lambda_1(R(\bm{a}))$. Let $v_{-2} \dots v_n$ denote the central path of $R(\bm{a})$, and let $v_c$ denote the coxswain attached to $v_0$. Let $\bm{x}\colon V(R(\bm{a})) \to \mathbb{R}$ be a unit eigenvector of $R(\bm{a})$ associated with $\lambda$. Clearly we have $\lambda x_c = x_0$, which implies that
    \begin{equation} \label{eqn:rowing-0}
        x_c^2 = \frac{x_c^2 + x_0^2}{1+\lambda^2} \le \frac{1}{1+\lambda^2}.
    \end{equation}
    Let $L$ be the rowing graph $R(\bm{a})$ with $v_c$ removed, and let $\bm{x}_L$ be $\bm{x}$ restricted to $V(L)$. Notice that $L$ is a line graph of a caterpillar tree, and so $\lambda_1(L) \ge -2$ (see \cref{footnote:line-graph}). Finally we bound the smallest eigenvalue of $R(\bm{a})$ as follows:
    \begin{multline*}
        \lambda = \bm{x}^\intercal A_{R(\bm{a})} \bm{x} = \bm{x}_L^\intercal A_L \bm{x}_L + 2x_cx_0 \ge -2\bm{x}_L^\intercal \bm{x}_L + 2x_cx_0 \\
        = -2(1-x_c^2) + 2\lambda x_c^2 = -2+(2+2\lambda)x_c^2 \stackrel{\eqref{eqn:rowing-0}}{\ge} -2 + \frac{2+2\lambda}{1+\lambda^2},
    \end{multline*}
    which implies that $\lambda \ge -1-\sqrt{2}$. In the above inequality, we assumed that $\lambda \le -1$ which follows from the fact that $R(\bm{a})$ has an edge.
\end{proof}

\begin{proof}[Proof of \ref{lem:rowing-a}]
    Let $\bm{x}$ be the vector that assigns $1$ to $v_{-2}$, $-2$ to $v_{-1}$, $4$ to $v_0$, $0$ to $v_1$, $-2$ to the coxswain, and $-4/a_1$ to every vertex in the clique of size $a_1$. The Rayleigh principle says that $\lambda_1(R(a_1))$ is at most
    \[
        \frac{\bm{x}^\intercal A_{R(a_1)} \bm{x}}{\bm{x}^\intercal \bm{x}} = \frac{-2(1 \cdot 2 + 2 \cdot 4 + 4 \cdot 2 + a_1 \cdot 4 \cdot (4/a_1)) + a_1(a_1-1) \cdot (-4/a_1)^2}{1^2 + (-2)^2 + 4^2 + (-2)^2 + a_1 \cdot (-4/a_1)^2},
    \]
    which approaches $-52/25 = -2.08$ as $a_1 \to \infty$.
\end{proof}

\begin{proof}[Proof of \ref{lem:rowing-b}]
    Take $\ell = \lceil 2/\varepsilon \rceil + 5$. Let $v_{-2} \dots v_{n_1 + \ell + n_2}$ denote the central path of the rowing graph $R(\bm{a}, 0^{(\ell)}, \bm{b})$, where $\bm{a} = (a_1, \dots, a_{n_1})$ and $\bm{b} = (b_1, \dots, b_{n_2})$, and let $\bm{x}\colon V(R(\bm{a},0^{(\ell)},\bm{b})) \to \mathbb{R}$ be a unit eigenvector of $R(\bm{a},0^{(\ell)},\bm{b})$ associated with the smallest eigenvalue. Choose $k \in \sset{0, \dots, \ell - 1}$ such that $x_{n_1+k}x_{n_1+k+1}$ reaches the minimum in absolute value. In particular, using the inequality $\abs{x_{n_1+i}x_{n_1+i+1}} \le (x_{n_1+i}^2 + x_{n_1+i+1}^2)/2$, we obtain
    \begin{equation} \label{eqn:rowing-b}
        \abs{x_{n_1+k}x_{n_1+k+1}} \le \frac{1}{\ell}\sum_{i = 0}^{\ell - 1}\abs{x_{n_1+i}x_{n_1+i+1}} \le \frac{1}{\ell} \sum_{i=0}^{\ell}x_{n_1+i}^2 \le \frac{1}{\ell} < \frac{\varepsilon}{2}.
    \end{equation}

    Notice that removing the edge $v_{n_1+k} v_{n_1+k+1}$ disconnects $R(\bm{a},0^{(\ell)},\bm{b})$ into two subgraphs, one of which is $R(\bm{a}, 0^{(k)})$, while the other is a line graph, denoted $L$, of a caterpillar tree. Clearly
    \begin{equation} \label{eqn:rowing-b-1}
        \lambda_1(R(\bm{a}, 0^{(k)})) \ge \lambda_1(R(\bm{a}, 0^{(\ell)})).
    \end{equation}
    As $\ell \ge 6$, $\widetilde{E}_8$ in \cref{fig:dynkin} is a proper subgraph of $R(\bm{a}, 0^{(\ell)})$, and so $\lambda_1(R(\bm{a}, 0^{(\ell)})) < -2$. Together with $\lambda_1(L) \ge -2$ (see \cref{footnote:line-graph}), we obtain
    \begin{equation} \label{eqn:rowing-b-2}
        \lambda_1(L) > \lambda_1(R(\bm{a}, 0^{(\ell)})).
    \end{equation}
    Let $\bm{x}_R$ and $\bm{x}_L$ be the unit eigenvector $\bm{x}$ restricted to $V(R(\bm{a},0^{(k)}))$ and $V(L)$. Finally we bound the smallest eigenvalue of $R(\bm{a},0^{(\ell)},\bm{b})$ as follows:
    \begin{align*}
        \lambda_1(R(\bm{a},0^{(\ell)},\bm{b})) & \stackrel{\phantom{(\ref{eqn:rowing-b},\ref{eqn:rowing-b-1},\ref{eqn:rowing-b-2})}}{=} \bm{x} A_{R(\bm{a},0^{(\ell)},\bm{b})}\bm{x} \\
        & \stackrel{\phantom{(\ref{eqn:rowing-b},\ref{eqn:rowing-b-1},\ref{eqn:rowing-b-2})}}{=} \bm{x}_R^\intercal A_{R(\bm{a},0^{(k)})} \bm{x}_R + 2x_{n_1+k}x_{n_1+k+1} + \bm{x}_L^\intercal A_L\bm{x}_L \\
        & \stackrel{\phantom{(\ref{eqn:rowing-b},\ref{eqn:rowing-b-1},\ref{eqn:rowing-b-2})}}{\ge} \lambda_1(R(\bm{a},0^{(k)})) \bm{x}_R^\intercal \bm{x}_R + 2x_{n_1+k}x_{n_1+k+1} + \lambda_1(L)\bm{x}_L^\intercal \bm{x}_L \\
        & \stackrel{(\ref{eqn:rowing-b},\ref{eqn:rowing-b-1},\ref{eqn:rowing-b-2})}{>} \lambda_1(R(\bm{a},0^{(\ell)}))(\bm{x}_R^\intercal \bm{x}_R + \bm{x}_L^\intercal \bm{x}_L) - \varepsilon \\
        & \stackrel{\phantom{(\ref{eqn:rowing-b},\ref{eqn:rowing-b-1},\ref{eqn:rowing-b-2})}}{=} \lambda_1(R(\bm{a}, 0^{(\ell)})) - \varepsilon. \qedhere
    \end{align*}
\end{proof}

\begin{proof}[Proof of \ref{lem:rowing-c}]
    Set $\bm{a} = (a_1, \dots, a_n, 0^{(\ell)})$ and $\lambda = \lambda_1(R(\bm{a}))$. Let $\bm{x}\colon V(R(\bm{a})) \to \mathbb{R}$ be an eigenvector of $R(\bm{a})$ associated with $\lambda$. Let $v_{-2}\dots v_{n + \ell}$ denote the central path of the rowing graph $R(\bm{a})$. In $R(\bm{a})$, let $K$ denote the clique of size $a_n$ attached to both $v_{n-1}$ and $v_n$, and pick an arbitrary vertex $u$ from $K$. We clearly have
    \begin{subequations} \label{eqn:rowing-c-0}
        \begin{align}
            \lambda x_u & = x_{n-1} + \sigma + x_n, \\
            \lambda x_n & = x_{n-1} + \sigma + x_u + x_{n+1},
        \end{align}
    \end{subequations}
    where $\sigma = \sum_{u' \in V(K) \setminus \sset{u}} x_{u'}$.The above identities imply that $\lambda(x_u - x_n) = x_n - x_u - x_{n+1}$, and so
    \begin{equation} \label{eqn:rowing-c-1}
        x_u = x_n - \frac{x_{n+1}}{1 + \lambda}.
    \end{equation}

    Let $R(\tilde{\ba})$ be the rowing graph obtained from $R(\bm{a})$ by removing $u$ and attaching a clique, denoted $\tilde{K}$, to both $v_n$ and $v_{n+1}$, where $\tilde{\ba} = (a_1, \dots, a_n-1,a_{n+1},0^{(\ell-1)})$, and the order $a_{n+1}$ of $\tilde{K}$ is chosen later. In particular, $V(R(\tilde{\ba})) = V(R(\bm{a})) \setminus \sset{u} \cup V(\tilde{K})$.

    We define a vector $\tilde{\bx}\colon V(R(\tilde{\ba}))\to\mathbb{R}$ as follows:
    \[
        \tilde{x}_v = \begin{cases}
            x_n + x_u & \text{if } v = v_n; \\
            -(x_n + x_u + x_{n+1})/a_{n+1} & \text{if } v \in V(\tilde{K}); \\
            x_v & \text{otherwise}.
        \end{cases}
    \]
    Because $\bm{x}$ is a nonzero vector, one can check that so is $\tilde{\bx}$. We compute
    \[
        \tilde{\bx}^\intercal \tilde{\bx} = \bm{x}^\intercal \bm{x} - x_n^2 - x_u^2 + (x_n + x_u)^2 + \frac{(x_n + x_u + x_{n+1})^2}{a_{n+1}}.
    \]
    Moreover we can compare $\tilde{\bx}^\intercal A_{R(\tilde{\ba})} \tilde{\bx}$ and $\bm{x}^\intercal A_{R(\bm{a})} \bm{x}$ as follows
    \begin{multline*}
        \tilde{\bx}^\intercal A_{R(\tilde{\ba})} \tilde{\bx} = \bm{x}^\intercal A_{R(\bm{a})} \bm{x} - 2x_u\left(x_{n-1} + \sigma + x_n\right) + 2x_u\left(x_{n-1} + \sigma + x_{n+1} \right) \\
        - 2(x_n + x_u + x_{n+1})^2 - a_{n+1}(a_{n+1}-1)\left(\frac{x_n + x_u + x_{n+1}}{a_{n+1}}\right)^2,
    \end{multline*}
    which simplifies via \eqref{eqn:rowing-c-0} to
    \[
        \tilde{\bx}^\intercal A_{R(\tilde{\ba})} \tilde{\bx} = \lambda \bm{x}^\intercal \bm{x} - 2\lambda x_u^2 + 2x_u (\lambda x_n - x_u) - \left(1 - \frac{1}{a_{n+1}}\right)(x_n + x_u + x_{n+1})^2.
    \]
    The Rayleigh principle says that $\lambda_1(R(\bm{a}))$ is at most
    \[
        \frac{\tilde{\bx}^\intercal A_{R(\tilde{\ba})} \tilde{\bx}}{\tilde{\bx}^\intercal \tilde{\bx}} = \frac{\lambda \bm{x}^\intercal \bm{x} - 2\lambda x_u^2 + 2x_u (\lambda x_n - x_u) - \left(1 - 1/a_{n+1}\right)(x_n + x_u + x_{n+1})^2}{\bm{x}^\intercal \bm{x} - x_n^2 - x_u^2 + (x_n + x_u)^2 + (x_n + x_u + x_{n+1})^2/a_{n+1}},
    \]
    which, as $a_{n+1} \to \infty$, approaches
    \[
        \frac{\lambda \bm{x}^\intercal \bm{x} - 2\lambda x_u^2 + 2x_u(\lambda x_n - x_u) - (x_n + x_u + x_{n+1})^2}{\bm{x}^\intercal \bm{x} - x_n^2 - x_u^2 + (x_n + x_u)^2}. \footnote{Here we assumed that the denominator in the limit is nonzero. Indeed, suppose on the contrary that $\bm{x}^\intercal \bm{x} = x_n^2 + x_u^2 - (x_n + x_u)^2 = -2x_nx_u$. Because $-2x_nx_u \le x_n^2 + x_u^2 \le x_n^2 + x_u^2 + x_{n+1}^2 \le \bm{x}^\intercal \bm{x}$, it must be the case that $x_n + x_u = 0$ and $x_{n+1} = 0$. In view of \eqref{eqn:rowing-c-1}, we have $x_n = x_u = 0$ and hence $\bm{x} = \bm{0}$, which is a contradiction.}
    \]

    It suffices to prove that
    \[
    \lambda \bm{x}^\intercal \bm{x} - 2\lambda x_u^2 + 2x_u(\lambda x_n - x_u) - (x_n + x_u + x_{n+1})^2 \le \lambda\left( \bm{x}^\intercal \bm{x} - x_n^2 - x_u^2 + (x_n + x_u)^2 \right),
    \]
    which is equivalent to
    \[
        (x_n+x_u+x_{n+1})^2 + 2(1+\lambda)x_u^2 \ge 0.
    \]
    Using \eqref{eqn:rowing-c-1}, we know that the left hand side of the last inequality equals
    \begin{multline*}
        \left(2x_n + \frac{\lambda}{1+\lambda}x_{n+1}\right)^2 + 2(\lambda+1)\left(x_n-\frac{x_{n+1}}{1+\lambda}\right)^2 \\
        = (2\lambda + 6)x_n^2 - \frac{4}{1+\lambda}x_nx_{n+1}+\left(1+\frac{1}{(1+\lambda)^2}\right)x_{n+1}^2.
    \end{multline*}
    From \cref{lem:rowing}\ref{lem:rowing-0}, we know that $\lambda \ge -1-\sqrt{2}$ in addition to the assumption that $\lambda \le -2$. One can check
    \[
        1 + \frac{1}{(1+\lambda)^2} \ge 2\lambda + 6 \ge \frac{4}{1+\lambda}\cdot \frac{1}{\lambda} > 0, \quad \text{for }\lambda \in [-1-\sqrt2,-2].
    \]

    We are left to prove $x_n^2 - \lambda x_nx_{n+1} + x_{n+1}^2 \ge 0$. In fact, we prove inductively that $x_i^2 - \lambda x_ix_{i+1} + x_{i+1}^2 \ge 0$ for $i \in \sset{n, \dots, n + \ell - 1}$. The base case where $i = n + \ell - 1$ follows immediately from $\lambda x_{n+\ell} = x_{n+\ell-1}$. For the inductive step, using $\lambda x_{i+1} = x_i + x_{i+2}$, we obtain
    \[
        x_i^2 - \lambda x_i x_{i+1} + x_{i+1}^2 = (\lambda x_{i+1}-x_{i+2})^2 - \lambda (\lambda x_{i+1}-x_{i+2})x_{i+1} + x_{i+1}^2 = x_{i+1}^2 - \lambda x_{i+1}x_{i+2} + x_{i+2}^2,
    \]
    which is nonnegative by the inductive hypothesis.
\end{proof}

\begin{proof}[Proof of \ref{lem:rowing-d}]
    Take $m = \lceil 2/\varepsilon \rceil + 1$. Let $v_{-2} \dots v_n$ denote the central path of the rowing graph $R(\bm{a})$, where $\bm{a} = (a_1, \dots, a_n)$, and let $\bm{x}\colon V(R(\bm{a})) \to \mathbb{R}$ be a unit eigenvector associated with the smallest eigenvalue of $R(\bm{a})$. Choose $k \in \sset{1, \dots, m}$ such that $x_k$ reaches the minimum in absolute value. In particular,
    \begin{equation} \label{eqn:rowing-d}
        x_k^2 \le \frac{1}{m} \sum_{i=1}^{m}x_i^2 \le \frac{1}{m} < \frac{\varepsilon}{2}.
    \end{equation}

    Let $v_{-2} \dots v_{k-1}v_{\tilde{k}} v_k \dots v_n$ denote the central path of $R(\tilde{\ba})$, where $\tilde{\ba} = (a_1, \dots, a_{k-1},0,a_k, \dots, a_n)$. We naturally view the vertex set of $R(\tilde{\ba})$ as $V(R(\bm{a})) \cup \{ v_{\tilde{k}}\}$, and we extend the unit eigenvector $\bm{x}\colon V(R(\bm{a})) \to \mathbb{R}$ to $\tilde{\bx}\colon V(R(\tilde{\ba})) \to \mathbb{R}$ by setting $\tilde{x}_{\tilde{k}} = x_k$. The Rayleigh principle says that $\lambda_1(R(\tilde{\ba}))$ is at most
    \begin{equation*}
        \frac{\tilde{\bx}^\intercal A_{R(\tilde{\ba})} \tilde{\bx}}{\tilde{\bx}^\intercal \tilde{\bx}} = \frac{\bm{x}^\intercal A_{R(\bm{a})} \bm{x} + 2x_k^2}{\bm{x}^\intercal \bm{x} + x_k^2} = \frac{\lambda_1(R(\bm{a})) + 2x_k^2}{1 + x_k^2} \stackrel{\eqref{eqn:rowing-d}}{<} \frac{\lambda_1(R(\bm{a})) + \varepsilon}{1 + \varepsilon/2}. \qedhere
    \end{equation*}
\end{proof}

We have all the ingredients to establish \cref{thm:main1-3}

\begin{proof}[Proof of \cref{thm:main1-3}]
    In view of \cref{thm:shearer}, we may assume that $\lambda \in (\lambda^*, \lambda')$. Fix $\varepsilon > 0$. We shall construct a rowing graph with smallest eigenvalue in $(-(1+\varepsilon/2)\lambda - \varepsilon, -\lambda)$. We may assume that no rowing graph has its smallest eigenvalue in $(-\lambda - \varepsilon, -\lambda)$ because otherwise we are done. Define
    \begin{gather*}
        S := \dset{(a_1, \dots, a_n) \in \mathbb{N}^n}{n \in \mathbb{N}^+ \text{ and } a_n > 0}, \\
        A := \dset{(a_1, \dots, a_n) \in S}{\lambda_1(R(a_1, \dots, a_n, 0^{(\ell)})) < -\lambda \text{ for some }\ell \in \mathbb{N}}.
    \end{gather*}

    \begin{claim*}
        For every $m \in \mathbb{N}^+$ there exists $(a_1, \dots, a_n) \in A$ with exactly $m$ nonzero entries such that
        \begin{equation} \label{eqn:main1-3-claim}
            (a_1, \dots, a_{k-1}, 0, a_k, \dots, a_n) \not\in A\text{ for every }k \in \sset{1, \dots, n}.
        \end{equation}
    \end{claim*}

    \begin{claimproof}[Proof of Claim]
        We order the elements in $S$ as follows:
        \begin{multline*}
            (a_1, \dots, a_{n_1}) \prec (b_1, \dots, b_{n_2}) \text{ if and only if}\\
            (a_1, \dots, a_{n_1}, 0^{(n_2)}) \text{ strictly precedes } (b_1, \dots, b_{n_2}, 0^{(n_1)}) \text{ lexicographically}.
        \end{multline*}
        Alternatively, $(a_1, \dots, a_{n_1}) \prec (b_1, \dots, b_{n_2})$ if and only if there exists $\delta > 0$ such that $\sum_{i=1}^{n_1} a_ix^i < \sum_{i=1}^{n_2} b_ix^i$ for every $x \in (0, \delta)$. We shall repeatedly use the fact that for every $\ell \in \mathbb{N}^+$ the set
        \[
            S_\ell := \dset{\bm{a} \in S}{\text{the length of }\bm{a} \text{ is at most }\ell}
        \]
        does not contain an infinite descending chain, hence $(S_\ell, \preceq)$ is well-founded, that is, every nonempty subset of $S_\ell$ has a minimal element.\footnote{Assuming the axiom of dependent choice, a weak form of the axiom of choice, $(S_\ell, \preceq)$ is well-founded if it contains no countable infinite descending chains.} This fact can be established by a simple induction on $\ell$.

        The ordering on $S$ particularly implies that $(a_1, \dots a_{k-1}, 0, a_k, \dots, a_n) \prec (a_1, \dots, a_n)$ when both sequences are in $S$ and $k \in \sset{1, \dots, n}$. Thus it suffices to construct, for every $m \in \mathbb{N}^+$, a minimal element $\bm{a}^{(m)}$ of $A_m$, where
        \[
            A_m := \dset{(a_1, \dots, a_n) \in A}{(a_1, \dots, a_n) \text{ has }m\text{ nonzero entries}}.
        \]
        Our inductive construction additionally requires that $\bm{a}^{(1)}, \bm{a}^{(2)}, \bm{a}^{(3)},\dots$ form a descending chain.

        Apply \cref{lem:rowing}\ref{lem:rowing-b} to obtain $\ell \in \mathbb{N}^+$ such that
        \begin{multline} \label{eqn:main1-3-l}
            \lambda_1(R(a_1, \dots, a_{n_1}, 0^{(\ell)}, b_1, \dots, b_{n_2})) > \lambda_1(R(a_1, \dots, a_{n_1}, 0^{(\ell)})) - \varepsilon \\
            \text{for every }n_1, n_2 \in \mathbb{N}, (a_1, \dots, a_{n_1}) \in \mathbb{N}^{n_1} \text{ and } (b_1, \dots, b_{n_2}) \in \mathbb{N}^{n_2}.
        \end{multline}
        In particular, when $n_1 = 0$, we have
        \[
            \lambda_1(R(0^{(\ell)}, b_1, \dots, b_{n_2})) > \lambda_1(R(0^{(\ell)})) - \varepsilon \text{ for every }(b_1, \dots, b_{n_2}) \in \mathbb{N}^{n_2}.
        \]

        For the base case where $m = 1$, note that $R(0^{(\ell)})$ is just $E_{2,\ell}$ defined in \cref{fig:e2n}, which satisfies, via \cref{lem:e2n}, that
        \[
            \lambda_1(R(0^{(\ell)})) = \lambda_1(E_{2,\ell}) > -\lambda^* > -\lambda.
        \]
        Since no rowing graph has its smallest eigenvalue in $(-\lambda - \varepsilon, -\lambda)$, the above two inequalities imply
        \[
            \lambda_1(R(0^{(\ell)}, b_1, \dots, b_{n_2})) \ge -\lambda \text{ for every }(b_1, \dots, b_{n_2}) \in \mathbb{N}^{n_2},
        \]
        which further implies that the length of every sequence in $A_1$ is at most $\ell$, that is, $A_1 \subseteq S_{\ell}$. Moreover, \cref{lem:rowing}\ref{lem:rowing-a} implies that $A_1$ is nonempty. Therefore there exists a minimal element of $A_1$.

        For the inductive step, suppose that $m \in \mathbb{N}^+$, and that $\bm{a}^{(1)} \succ \dots \succ \bm{a}^{(m)}$ are minimal elements of $A_1, \dots, A_m$ respectively. Say $\bm{a}^{(m)} = (b_1, \dots, b_n)$. \cref{lem:rowing}\ref{lem:rowing-c} shows that there exists $b_{n+1} \in \mathbb{N}^+$ such that $\bm{b}^{(m+1)} := (b_1, \dots, b_{n-1}, b_n-1, b_{n+1}) \in A$. Since $\bm{b}^{(m+1)} \prec \bm{a}^{(m)}$ and $\bm{a}^{(m)}$ is minimal in $A_m$, it must be the case that $b_n > 1$ or equivalently $\bm{b}^{(m+1)} \in A_{m+1}$. It suffices to show that
        \[
            \dset{\bm{b} \in A_{m+1}}{\bm{b} \preceq \bm{b}^{(m+1)}} \subseteq S_{m\ell}.
        \]
        Assume for the sake of contradiction that there exists $\bm{b} \in A_{m+1}$ such that $\bm{b} \preceq \bm{b}^{(m+1)}$ and the length of $\bm{b}$ is more than $m\ell$. By the pigeonhole principle, there exists an initial segment, denoted $\bm{b}'$, of $\bm{b}$ such that $(\bm{b}', 0^{(\ell)})$ is an initial segment of $\bm{b}$. We may require in addition that either $\bm{b}'$ is the empty sequence or the last entry of $\bm{b}'$ is positive. If $\bm{b}'$ is the empty sequence, then we can argue similarly to the base case that
        \[
            \lambda_1(R(\bm{b}, c_1, \dots, c_{n'})) \ge -\lambda \text{ for every }(c_1, \dots, c_{n'}) \in \mathbb{N}^{n'},
        \]
        which contradicts with $\bm{b} \in A_{m+1}$.

        We may assume that the last entry of $\bm{b}'$ is positive. Let $i \in \sset{1, \dots, m}$ be the number of nonzero entries in $\bm{b}'$. Since $\bm{b}' \prec \bm{b} \preceq \bm{b}^{(m+1)} \prec \bm{a}^{(m)} \preceq \bm{a}^{(i)}$, and $\bm{a}^{(i)}$ is minimal in $A_i$, we know that $\bm{b}' \not\in A$, and in particular
        \[
            \lambda_1(R(\bm{b}', 0^{(\ell)})) \ge -\lambda.
        \]
        From \eqref{eqn:main1-3-l}, we know that
        \[
            \lambda_1(R(\bm{b}, c_1, \dots, c_{n'})) > \lambda_1(R(\bm{b}', 0^{(\ell)})) - \varepsilon \text{ for every }(c_1, \dots, c_{n'}) \in \mathbb{N}^{n'}.
        \]
        Since no rowing graph has its smallest eigenvalue in $(-\lambda-\varepsilon, -\lambda)$, the above two inequalities imply
        \[
            \lambda_1(R(\bm{b}, c_1, \dots, c_{n'})) \ge -\lambda \text{ for every }(c_1, \dots, c_{n'}) \in \mathbb{N}^{n'},
        \]
        which contradicts with $\bm{b} \in A_{m+1}$.
    \end{claimproof}

    Finally let $m$ be given by \cref{lem:rowing}\ref{lem:rowing-d}. The claim provides $(a_1, \dots, a_n) \in A$ with $m$ nonzero entries such that \eqref{eqn:main1-3-claim} holds. Let $\ell \in \mathbb{N}$ be such that
    \[
        \lambda_1(R(a_1, \dots, a_n, 0^{(\ell)})) < -\lambda.
    \]
    Since the length of $(a_1, \dots, a_n, 0^{(\ell)})$ is at least $m$, \cref{lem:rowing}\ref{lem:rowing-d} says that there exists $k \in \sset{1, \dots, m}$ such that
    \[
        \lambda_1(R(a_1, \dots, a_{k-1}, 0, a_k, \dots, a_n, 0^{(\ell)})) < \frac{\lambda_1(R(a_1, \dots, a_n, 0^{(\ell)})) + \varepsilon}{1 + \varepsilon/2},
    \]
    However \eqref{eqn:main1-3-claim} asserts that $(a_1, \dots, a_{k-1}, 0, a_k, \dots, a_n) \not\in A$, which implies that
    \[
        \lambda_1(R(a_1, \dots, a_{k-1}, 0, a_k, \dots, a_n, 0^{(\ell)})) \ge -\lambda.
    \]
    Combining the last three inequalities, we obtain that
    \begin{equation*}
        -(1 + \varepsilon/2)\lambda - \varepsilon < \lambda_1(R(a_1,\dots, a_n, 0^{(\ell)})) < -\lambda. \qedhere
    \end{equation*}
\end{proof}

\section{Forbidden subgraphs for $\G^\pm(\la)$} \label{sec:forb-signed}

A useful tool in spectral graph theory for signed graphs is switching --- two signed graphs are \emph{switching equivalent} if one graph can be obtained from the other by negating all the edges in a cut-set. An important feature of switching equivalence is that the switching equivalent signed graphs all have the same spectrum.

To prove the second main theorem, we need to extend the concepts and results from the previous section. We recommend that the reader go through the rest of the section alongside \cref{sec:forb}.

\subsection{Proof of \cref{thm:main2} for $\lambda < 2$}

We first generalize path extension, path-clique extension, clique extension, and \cref{lem:hoffman}(\ref{lem:c1}, \ref{lem:c2}).

\begin{definition}
    Given a nonempty vertex subset $A$ of a signed graph $F^\pm$, $\ell \in \mathbb{N}$, and $m \in \mathbb{N}^+$, a \emph{signed vertex subset} $A^\pm$ of $F^\pm$ is the vertex subset $A$ together with an assignment of signs (positive or negative);
    \begin{enumerate}[nosep, label=(\alph*)]
        \item the \emph{path extension} $(F^\pm, A^\pm, \ell)$ is obtained from $F^\pm$ by adding an all-positive path $v_0 \dots v_\ell$ of length $\ell$, and connecting $v_0$ to every vertex $v$ in $A$ by an edge signed according to the sign of $v$ in $A^\pm$;
        \item the \emph{path-clique extension} $(F^\pm, A^\pm, \ell, K_m)$ is further obtained from $(F^\pm, A^\pm, \ell)$ be adding an all-positive clique of size $m$, and connecting every vertex in the clique to $v_\ell$ by a positive edge;
        \item the \emph{clique extension} $(F^\pm, A^\pm, K_m)$ is obtained from $F^\pm$ by adding a clique of size $m$, and connecting every vertex in the clique to every vertex $v$ in $A$ by an edge signed according to the sign of $v$ in $A^\pm$.
    \end{enumerate}
\end{definition}

\begin{lemma} \label{lem:hoffman-signed}
    For every signed cycle $C^\pm_n$ of length $n$, and every signed set $A^\pm$ of two adjacent vertices of $C^\pm_n$, the smallest eigenvalues of its extensions satisfy:
    \begin{itemize}[nosep]
        \item[(c1)] $\lim_m \lambda_1(C_n^\pm, A^\pm, \ell, K_m) \le -2$ for fixed $\ell \in \mathbb{N}$, and \wlabel{lem:c1s}{c1}
        \item[(c2)] $\lim_m \lambda_1(C_n^\pm, A^\pm, K_m) \le -2$.\wlabel{lem:c2s}{c2}
    \end{itemize}
\end{lemma}

\begin{proof}
    Let $v_0,v_1,\dots, v_{n-1}$ be the vertices of $C_n^\pm$, and let $\sigma\colon E(C^\pm_n) \to \sset{\pm 1}$ be the signing of $C^\pm_n$. Suppose that $A^\pm$ is a signed set of $\sset{v_0,v_{n-1}}$. By switching a subset of $\sset{v_0, v_{n-1}}$, we may assume that $A^\pm = \{v_0^+, v_{n-1}^+\}$. If $\sigma_{0,n-1} = -1$, then both $(C_n^\pm, A^\pm, \ell, K_m)$ and $(C_n^\pm, A^\pm, K_m)$ contain a subgraph that is switching equivalent to the all-negative triangle $-K_3$, and so their smallest eigenvalues are at most $\lambda_1(-K_3) = -2$. Hereafter we assume that $\sigma_{0,n-1} = +1$. Furthermore, we may assume that $C_n^\pm$ has an odd number of positive edges. Indeed, if $C_n^\pm$ has an even number of positive edges, then $C^\pm_n$ is already switching equivalent to the all-negative cycle of length $n$, whose smallest eigenvalue equals $-2$, and so we are done again.

    For (\ref{lem:c1s}), we define a vector $\bm{x}\colon V(C_n^\pm, A^\pm, K_m) \to \mathbb{R}$ as follows. Let $v_{n}v_{n+1}\dots v_{n+\ell}$ and $K_m$ be the path and the clique added to $C_n^\pm$ to obtain the path-clique extension $(C_n^\pm, A^\pm, \ell, K_m)$, where $v_{n}$ is adjacent to $v_0$ and $v_{n-1}$, and $v_{n+\ell}$ is adjacent to every vertex in $K_m$. Assign $x_0 = x_{n-1} = -1$, $x_{n+i} = 2(-1)^i$ for $i \in \sset{0, \dots, \ell}$ and $x_u = -x_{n+\ell}/m$ for every $u \in V(K_m)$. Since an even number of edges among $v_0v_1, v_1v_2, \dots, v_{n-2}v_{n-1}$ are positive in $C_n^\pm$, we may choose $x_1, \dots, x_{n-2}\in \sset{\pm 1}$ such that $x_ix_{i+1} = -\sigma_{i,i+1}$ for every $i \in \sset{0, \dots, n-2}$. The Rayleigh principle says that the smallest eigenvalue of $(C_n^\pm, A^\pm, \ell, K_m)$ is at most
    \begin{multline*}
        \bigg(2\sum_{i=0}^{n-2}\sigma_{i,i+1}x_ix_{i+1} + 2x_0x_{n-1} + 2(x_0 + x_{n-1})x_n + 2\sum_{i=0}^{\ell-1}x_{n+i}x_{n+i+1} - 2x_{n+\ell}^2 + \\
        + m(m-1)(x_{n+\ell}/m)^2\bigg) \bigg/ \bigg(\sum_{i=0}^{n-1} x_i^2 + \sum_{i=0}^\ell x_{n+i}^2 + m(x_{n+\ell}/m)^2\bigg),
    \end{multline*}
    which equals $(-2n - 8(\ell+1) - 4/m)/(n + 4(\ell+1) + 4/m)$, and approaches $-2$ as $m \to \infty$.

    For (\ref{lem:c2s}), we define a vector $\bm{x} \colon V(C_n^\pm, A^\pm, \ell, K_m) \to \mathbb{R}$ as follows. Let $K_m$ be the clique added to $C_n^\pm$ to obtain the clique extension $(C_n^\pm, A^\pm, K_m)$. Assign $x_0 = x_{n-1} = -1$ and $x_u = -(x_0 + x_{n-1})/m$ for every $u \in V(K_m)$. Similar to (\ref{lem:c1s}), we may choose $x_1, \dots, x_{n-2}\in \sset{\pm 1}$ such that $x_ix_{i+1} = -\sigma_{i,i+1}$ for every $i \in \sset{0, \dots, n-2}$. The Rayleigh principle says that the smallest eigenvalue of $(C_n^\pm, A^\pm, K_m)$ is at most
    \[
        \frac{2\sum_{i=0}^{n-2}\sigma_{i,i+1}x_ix_{i+1} + 2x_0x_{n-1} - 2(x_0 + x_{n-1})^2 + m(m-1)((x_0 + x_{n-1})/m)^2}{\sum_{i=0}^{n-1} x_i^2 + m((x_0 + x_{n-1})/m)^2},
    \]
    which equals $(-2n - 4/m)/(n + 4/m)$, and approaches $-2$ as $m \to \infty$.
\end{proof}

We naturally generalize extension family as well as \cref{lem:hoffman-extension}.

\begin{definition}[Extension family]
    For every signed graph $F^\pm$ and $\ell, m \in \mathbb{N}^+$, the \emph{extension family} $\mathcal{X}^\pm(F^\pm, \ell, m)$ of $F^\pm$ consists of the path-extension $(F^\pm, A^\pm, \ell)$, the path-clique extensions $(F^\pm, A^\pm, \ell_0, K_m)$, where $\ell_0 \in \sset{0, \dots, \ell-1}$, and the clique extension $(F^\pm, A^\pm, K_m)$, for every nonempty signed vertex subset $A^\pm$ of $F^\pm$.
\end{definition}

\begin{lemma} \label{lem:hoffman-extension-signed}
    For every $\lambda < 2$, there exist $\ell, m \in \mathbb{N}^+$ such that both the extension family $\mathcal{X}^\pm(C, \ell, m)$ of the claw graph $C$, and the extension family $\mathcal{X}^\pm(D, \ell, m)$ of the diamond graph $D$ are disjoint from $\G^\pm(\la)$.
\end{lemma}

\begin{proof}
    From \cref{lem:hoffman}(\ref{lem:n1}, \ref{lem:n2}, \ref{lem:p}) and \cref{lem:hoffman-extension}, we obtain $\ell, m \in \mathbb{N}^+$ such that $P_\ell \not\in \G^\pm(\la)$, none of the following graphs
    \begin{equation} \label{eqn:signed-k2-extensions}
        (\overline{K}_2, V(\overline{K}_2), 0, K_m), \dots, (\overline{K}_2, V(\overline{K}_2), \ell-1, K_m), \text{ and }(\overline{K}_2, V(\overline{K}_2), K_m)
    \end{equation}
    is in $\G^\pm(\la)$, and $\mathcal{X}(C, \ell, m) \cup \mathcal{X}(D, \ell, m)$ is disjoint from $\G^\pm(\la)$. Since $P_\ell \not\in \G^\pm(\la)$, no path extension in $\mathcal{X}^\pm(C, \ell, m) \cup \mathcal{X}^\pm(D, \ell, m)$ is in $\G^\pm(\la)$.

    We are left to deal with the path-clique extensions and the clique extension
    \begin{equation} \label{eqn:signed-extension}
        (F, A^\pm, 0, K_m), \dots, (F, A^\pm, \ell-1, K_m), \text{ and }(F, A^\pm, K_m),
    \end{equation}
    where $F$ is $C$ or $D$, and $A^\pm$ is the signed set of a nonempty vertex subset $A$ of $F$. We break the rest of the proof into three cases.

    \paragraph{Case 1:} $A^\pm$ is all-positive or all-negative. Since that the signed graphs in \eqref{eqn:signed-extension}, after switching the cut-set between $V(F)$ and its complement in case $A^\pm$ is all-negative, are respectively $(F, A, 0, K_m), \dots, (F, A, \ell-1, K_m)$, and $(F, A, K_m)$, none of them is in $\G^\pm(\la)$.

    \paragraph{Case 2:} There exists an edge $uv$ of $F$ such that $\sset{u^-, v^+} \subseteq A^\pm$. Since the all-negative triangle $-K_3$, whose smallest eigenvalue equals $-2$, is switching equivalent to a subgraph of $(F, A^\pm, 0)$, no signed graph in \eqref{eqn:signed-extension} is in $\G^\pm(\la)$.

    \paragraph{Case 3:} $A^\pm$ is neither all-positive nor all-negative, and no edge $uv$ of $F$ satisfies that $\sset{u^-, v^+} \subseteq A^\pm$. Label the vertices of $C$ and $D$ as in \cref{fig:claw-and-diamond}. We may assume without loss of generality that $\sset{1^-, 3^+} \subseteq A^\pm$. One can check that, for $F \in \sset{C, D}$, the signed graphs in \eqref{eqn:signed-k2-extensions}, after switching at the vertex labeled by $1$, are respectively subgraphs of those in \eqref{eqn:signed-extension}, and hence no signed graph in \eqref{eqn:signed-extension} is in $\G^\pm(\la)$.
\end{proof}

Next, generalizing \cref{lem:forbid-extensions}, we show that forbidding a star, an all-negative complete graph, and an extension family of $F^\pm$ effectively forbids $F^\pm$ itself in every sufficiently large connected signed graph.

\begin{lemma} \label{lem:forbid-extensions-signed}
    For every signed graph $F^\pm$ and $k_1, k_2, \ell, m \in \mathbb{N}^+$, there exists $N \in \mathbb{N}$ such that for every connected signed graph $G^\pm$ with more than $N$ vertices, if $G^\pm$ does not contain any subgraph that is switching equivalent to any member in $\sset{S_{k_1}, -K_{k_2}} \cup \mathcal{X}^\pm(F^\pm, \ell, m)$, then $G^\pm$ does not contain any subgraph that is switching equivalent to $F^\pm$ either. \qed
\end{lemma}

We leave the proof to the reader as one can make the proof of \cref{lem:forbid-extensions} into that for \cref{lem:forbid-extensions-signed} by \emph{mutatis mutandis}. To get started, one might want to take $N = d^{\ell + v}$, where $d$ is the Ramsey number $R(k_1, k_2, 3^{v+1} m + v)$.

The last ingredient is a sufficient condition for signed line graphs, which generalizes \cref{thm:claw-diamond}.

\begin{definition}[Bidirected graph and signed line graph] \label{def:signed-line-graph}
    A \emph{bidirected graph} is a graph in which each vertex-edge incidence has a positive or negative sign. We decorate variables for bidirected graphs with an arrow on the top. A signed graph $G^\pm$ is the \emph{signed line graph} of a bidirected graph $\vec{H}$ if the underlying graph of $G^\pm$ is the line graph of the underlying graph of $\vec{H}$, and moreover for every two distinct edges $e_1$ and $e_2$ of $\vec{H}$ having a vertex $v$ in common, the sign of the edge $e_1 e_2$ in $G^\pm$ is the product of the signs of the two incidences $(v, e_1)$ and $(v, e_2)$ in $\vec{H}$.
\end{definition}

Pictorially we place a sign close to each incidence in a bidirected graph. See \cref{fig:bidirected-graph} for an example of a bidirected graph and its signed line graph.

\begin{figure}
    \centering

    \begin{tikzpicture}[
        very thick, scale=0.75, baseline=(v.base),
        neg/.pic = {\draw[fill=white] (0,0) circle (0.1); \draw (-0.1,0)--(0.1,0);},
        pos/.pic = {\draw[fill=white] (0,0) circle (0.1); \draw (-0.1,0)--(0.1,0); \draw (0,0.1)--(0,-0.1);},
    ]
        \coordinate (v) at (0,0);
        \draw (-2,0) -- (0,1) -- (2,0) -- (0,-1) -- (-2,0) -- (2,0);
        \draw (-1.4,0.3) pic {neg};
        \draw (-0.4,0.8) pic {pos};
        \draw (0.4,0.8) pic {pos};
        \draw (1.4,0.3) pic {pos};
        \draw (-1.4,-0.3) pic {neg};
        \draw (-0.4,-0.8) pic {pos};
        \draw (1.4,-0.3) pic {pos};
        \draw (0.4,-0.8) pic {neg};
        \draw (-1.1,0) pic {neg};
        \draw (1.1,0) pic {neg};
        \node[vertex] at (-2,0) {};
        \node[vertex] at (2,0) {};
        \node[vertex] at (0,-1) {};
        \node[vertex] at (0,1) {};
        \draw (-2,0) node[left]{$v_1$};
        \draw (0,1) node[above]{$v_2$};
        \draw (0,-1.1) node[below]{$v_3$};
        \draw (2,0) node[right]{$v_4$};
    \end{tikzpicture}\qquad
    \begin{tikzpicture}[very thick, scale=0.5, baseline=(v.base)]
        \coordinate (v) at (0,0);
        \draw (-3,1.5) -- (3,1.5) -- (3,-1.5);
        \draw[dashed] (3,-1.5) -- (-3,-1.5);
        \draw (-3,-1.5) -- (-3,1.5) -- (0,0);
        \draw[dashed] (0,0) -- (3,-1.5);
        \draw (-3,-1.5) -- (0,0);
        \draw[dashed] (0,0) -- (3,1.5);
        \node[vertex] at (-3,1.5) {};
        \node[vertex] at (3,-1.5) {};
        \node[vertex] at (-3,-1.5) {};
        \node[vertex] at (3,1.5) {};
        \node[vertex] at (0,0) {};
        \node[below] at (0,-0.3) {$v_1v_4$};
        \node[above] at (-3.5,1.5) {$v_1v_2$};
        \node[above] at (3.5,1.5) {$v_2v_4$};
        \node[below] at (-3.5,-1.6) {$v_1v_3$};
        \node[below] at (3.5,-1.6) {$v_3v_4$};
    \end{tikzpicture}
    \caption{A bidirected graph and its signed line graph. In a signed graph, the positive edges are represented by solid segments and the negative edges are represented by dashed segments.} \label{fig:bidirected-graph}
\end{figure}

\begin{remark}
    Our definition of a signed line graph is different from the existing literature --- the conventional definition (e.g.\ Zaslavsky~\cite{Z10}) negates all the edge signs of $G^\pm$ in \cref{def:signed-line-graph}. Our choice of the definition is based on aesthetic reasons: the signed line graph of an all-positive bidirected graph is still all-positive, and the smallest eigenvalue of a signed line graph is at least $-2$.
\end{remark}

\begin{lemma}[Proposition~2.2 of Vijayakumar~\cite{V87}] \label{lem:claw-diamond-signed}
    For every signed graph $G^\pm$, if $G^\pm$ does not contain any subgraph that is switching equivalent to the claw graph, or the diamond graph, or the all-negative triangle, then $G^\pm$ is a signed line graph.
\end{lemma}

Although the conclusion in \cite[Proposition~2.2]{V87} is weaker than that in \cref{lem:claw-diamond-signed}, Vijayakumar's proof already gives \cref{lem:claw-diamond-signed}. Below we reproduce his proof in terms of signed line graph.

\begin{proof}
    Let $G$ be the underlying graph of $G^\pm$. One can check that $G$ contains neither the claw graph nor the diamond graph as a subgraph. Thus \cref{thm:claw-diamond} implies that $G$ is a line graph of another graph, denoted $H$, without isolated vertices. We may identify the vertex set of $G^\pm$ with the edge set of $H$.

    For every $v \in V(H)$, let $E_v(H)$ be the set of edges that are incident to $v$ in $H$. Define a signing $\sigma \colon \dset{(v, e)}{v \in V(H), e \in E_v(H)} \to \sset{\pm1}$ such that for every $v \in V(H)$,
    \begin{equation} \label{eqn:bidirected-graph}
        \sigma(v, e_1)\sigma(v, e_2) = A_{G^\pm}(e_1,e_2) \text{ for distinct } e_1, e_2 \in E_v(H).
    \end{equation}
    Such a signing $\sigma$ can be chosen as follows: for every $v\in V(H)$, define $\sigma(v, e_0) = 1$ for some arbitrary $e_0 \in E_v(H)$; and define $\sigma(v, e) = A_{G^\pm}(e_0, e)$ for any other $e \in E_v(H)$. Since $G^\pm$ does not contain any subgraph that is switching equivalent to the all-negative triangle, it can be seen that \eqref{eqn:bidirected-graph} holds. Assigning $\sigma(v, e)$ to every incidence $(v, e)$ of $H$, we obtain a bidirected graph $\vec{H}$ whose signed line graph is $G^\pm$.
\end{proof}

We are ready to present the proof of the second main theorem for $\lambda < 2$.

\begin{proof}[Proof of \cref{thm:main2} for $\lambda < 2$]
    Denote $\mathcal{C}^\pm_n$ the set of signed cycles of length $n$. We obtain $\ell, m \in \mathbb{N}^+$ from \cref{lem:hoffman,lem:hoffman-signed,lem:hoffman-extension-signed} such that the following family
    \begin{align*}
        \widetilde{\mathcal{F}}_1^\pm := & \sset{S_4, P_\ell, -K_3} \cup \mathcal{X}^\pm(C, \ell, m) \cup \mathcal{X}^\pm(D, \ell, m) \\
        & \cup \dset{(C_n^\pm, V_2(C_n), \ell_0, K_m)}{n \in \sset{3, \dots, \ell + 1}, C_n^\pm \in \mathcal{C}_n^\pm, \ell_0 \in \sset{0, \dots, \ell - 1}} \\
        & \cup \dset{(C_n^\pm, V_2(C_n), K_m)}{n \in \sset{3, \dots, \ell + 1}, C_n^\pm \in \mathcal{C}_n^\pm} \\
        & \cup \dset{(K_m, V(K_m), \ell_0, K_m)}{\ell_0 \in \sset{0, \dots, \ell-1}}
    \end{align*}
    is disjoint from $\G^\pm(\la)$. Let $\mathcal{F}_1^\pm$ be the smallest family, that is closed under switching, containing $\widetilde{\mathcal{F}}_1^\pm$. Clearly $\mathcal{F}_1^\pm$ is also disjoint from $\G^\pm(\la)$.

    We claim that there exists $N \in \mathbb{N}$ such that for every connected signed graph $G^\pm$ with more than $N$ vertices, if $G^\pm$ contains no member in $\mathcal{F}_1^\pm$ as a subgraph, then there exists a bidirected rooted tree $\vec{H} \in \mathcal{T}_N$ such that $G^\pm$ is the signed line graph of $\vec{H}$, where $\mathcal{T}_N$ is the family of bidirected rooted tree such that every connected component obtained from removing the root has at most $N$ vertices. We omit the proof of the claim as it, \emph{mutatis mutandis}, is very much like the one in the proof of \cref{thm:main1} for $\lambda < 2$ towards the end of \cref{subsec:less-than-2}.

    Let $N$ be given by the above claim, and set
    \begin{align*}
        \mathcal{F}_0^\pm & := \dset{G^\pm \not\in \G^\pm(\la)}{G^\pm \text{ has at most }N\text{ vertices}}, \\
        \widetilde{\mathcal{F}}_2^\pm & := \dset{G^\pm \not\in \G^\pm(\la)}{\exists \vec{H} \in \mathcal{T}_N \text{ s.t. }G^\pm \text{ is the signed line graph of }\vec{H}}.
    \end{align*}
    Setting $\mathcal{F}_2^\pm$ the family of signed graphs that are minimal in $\widetilde{\mathcal{F}}_2^\pm$ under taking subgraphs, one can check that $\mathcal{F}_0^\pm \cup \mathcal{F}_1^\pm \cup \mathcal{F}_2^\pm$ is a forbidden subgraph characterization of $\G^\pm(\la)$.

    It suffices to prove that $\mathcal{F}_2^\pm$ is finite. Let $\vec{T}_1, \dots, \vec{T}_n$ be an enumeration of bidirected rooted trees $\vec{T}$ on at most $N+1$ vertices. We encode $G^\pm \in \mathcal{F}_2^\pm$ by $t_{G^\pm} \in \mathbb{N}^n$ as follows. Let $\vec{H}$ be the bidirected root tree in $\mathcal{T}_N$ such that $G^\pm$ is the signed line graph of $\vec{H}$. After removing the root $u$ from $\vec{H}$, let $U_1, U_2, \dots, U_m$ be the vertex sets of the connected components. For $i \in \sset{1, \dots, m}$, we view the subgraph $\vec{H}_i := \vec{H}[\sset{u} \cup U_i]$ as a bidirected tree rooted at $u$. Set $t_{G^\pm} = (t_1, \dots, t_n)$, where $t_i$ is the number of occurrences of $\vec{T}_i$ in $\vec{H}_1, \dots, \vec{H}_m$. Because no member of $\mathcal{F}_2^\pm$ is a subgraph of any other, one can deduce that $\dset{t_{G^\pm}}{G^\pm \in \mathcal{F}_2^\pm}$ is an antichain in $(\mathbb{N}^n, \le)$, and so $\mathcal{F}_2^\pm$ is finite by \cref{lem:dickson}.
\end{proof}

\subsection{Proof of \cref{thm:main2} for $\lambda \in [2, \lambda^*)$}

We prove a generalization of \cref{thm:main1-2}, from which the second main theorem for $\lambda \in [2, \lambda^*)$ follows.

\begin{theorem} \label{thm:main2-2}
    For every $\lambda \in [2,\lambda^*)$, the number of connected signed graphs in $\G^\pm(\la) \setminus \mathcal{G}^\pm(2)$ is finite.
\end{theorem}

\begin{proof}[Proof of \cref{thm:main2} for $\lambda \in [2, \lambda^*)$]
    Using the fact that every minimal forbidden subgraph for $\mathcal{G}^\pm(2)$ has at most $10$ vertices (see \cite{V87}), and \cref{thm:main2-2}, one can prove this case by following the argument in the proof of \cref{thm:main1} for $\lambda \in [2, \lambda^*)$ in \cref{sec:main1-2}.
\end{proof}

Recall from \cref{thm:cgss} that a graph is a generalized line graph if and only if it is represented by a subset of the root system $D_n$ defined in \cref{def:dn-e8}. For signed graphs, we work directly with those that are represented by a subset of $D_n$.

\begin{definition}[Representation of signed graphs]
    Given a signed graph $G^\pm$ and a subset $V$ of $\mathbb{R}^n$, we say that $G^\pm$ is \emph{represented} by $V$ if the Gram matrix of $V$ equals $A_{G^\pm} + 2I$, where $A_{G^\pm}$ is the signed adjacency matrix of $G^\pm$.
\end{definition}

These signed graphs that are represented by a subset of $D_n$, like generalized line graphs, also have a finite forbidden subgraph characterization.

\begin{theorem}[Chawathe and Vijayakumar~\cite{CV90}]
    The minimal forbidden subgraphs for the family $\mathcal{D}_\infty^\pm$ of signed graphs, that are represented by a subset of $D_n$, are listed in \cref{fig:minimal-forb-1,fig:minimal-forb-2} up to switching equivalence. \qed
\end{theorem}

\begin{figure}
    \centering
    \begin{tikzpicture}[very thick, baseline=(v.base)]
    \coordinate (v) at (0,.25);
    \coordinate (v0) at (0,0);
    \path (v0) + (90:1) coordinate (v1);
    \path (v0) + (210:1) coordinate (v2);
    \path (v0) + (330:1) coordinate (v3);
    \draw (v0) -- (v1);
    \draw (v2) -- (v1);
    \draw (v3) -- (v1);
    \draw[dashed] (v2) -- (v3);
    \node[circ] at (v0) {0};
    \node[circ] at (v2) {1};
    \node[circ] at (v3) {2};
    \node[circ] at (v1) {3};
    \node at (0,-1.25) {$S_{32}$};
    \end{tikzpicture}
    \begin{tikzpicture}[very thick, baseline=(v.base)]
    \coordinate (v) at (0,.25);
    \coordinate (v0) at (0,0);
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    \draw (v0) -- (v2);
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    \node[circ] at (v2) {2};
    \node[circ] at (v3) {3};
    \node at (0,-1.25) {$S_{33}$};
    \end{tikzpicture}
    \begin{tikzpicture}[very thick, baseline=(v.base)]
    \coordinate (v) at (0,.25);
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    \node[circ] at (v1) {1};
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    \node[circ] at (v3) {3};
    \node at (0,-1.25) {$S_{34}$};
    \end{tikzpicture}
    \begin{tikzpicture}[very thick, baseline=(v.base)]
    \coordinate (v) at (0,.25);
    \coordinate (v0) at (0,0);
    \path (v0) + (90:1) coordinate (v1);
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    \draw (v1) -- (v3);
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    \node[circ] at (v2) {2};
    \node[circ] at (v3) {3};
    \node at (0,-1.25) {$S_{35}$};
    \end{tikzpicture}
    \begin{tikzpicture}[very thick, baseline=(v.base)]
    \coordinate (v) at (0,.1);
    \coordinate (v0) at (0,0);
    \path (v0) + (90:1) coordinate (v1);
    \path (v0) + (162:1) coordinate (v2);
    \path (v0) + (234:1) coordinate (v3);
    \path (v0) + (306:1) coordinate (v4);
    \path (v0) + (378:1) coordinate (v5);
    \draw (v0) -- (v1);
    \draw (v3) -- (v2);
    \draw (v4) -- (v5);
    \draw (v5) -- (v1);
    \draw (v2) -- (v1);
    \draw[dashed] (v3) -- (v4);
    \node[circ] at (v0) {0};
    \node[circ] at (v3) {1};
    \node[circ] at (v4) {2};
    \node[circ] at (v5) {3};
    \node[circ] at (v2) {4};
    \node[circ] at (v1) {5};
    \node at (0,-1.4) {$S_{36}$};
    \end{tikzpicture}
    \begin{tikzpicture}[very thick, baseline=(v.base)]
    \coordinate (v) at (0,.1);
    \coordinate (v0) at (0,0);
    \path (v0) + (90:1) coordinate (v1);
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    \draw (v2) -- (v3);
    \draw (v2) -- (v1);
    \draw (v5) -- (v4);
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    \draw (v0) -- (v3);
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    \node[circ] at (v3) {4};
    \node[circ] at (v1) {5};
    \node at (0,-1.4) {$S_{37}$};
    \end{tikzpicture}
    \begin{tikzpicture}[very thick, baseline=(v.base)]
    \coordinate (v) at (0,.1);
    \coordinate (v0) at (0,0);
    \path (v0) + (90:1) coordinate (v1);
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    \path (v0) + (378:1) coordinate (v5);
    \draw (v4) -- (v5);
    \draw (v4) -- (v0);
    \draw (v3) -- (v2);
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    \node[circ] at (v1) {4};
    \node[circ] at (v0) {5};
    \node at (0,-1.4) {$S_{38}$};
    \end{tikzpicture}
    \begin{tikzpicture}[very thick, baseline=(v.base)]
    \coordinate (v) at (0,.1);
    \coordinate (v0) at (0,0);
    \path (v0) + (90:1) coordinate (v1);
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    \draw (v1) -- (v0);
    \draw (v2) -- (v3);
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    \node[circ] at (v2) {1};
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    \node[circ] at (v4) {4};
    \node[circ] at (v0) {5};
    \node at (0,-1.4) {$S_{39}$};
    \end{tikzpicture}
    \begin{tikzpicture}[very thick, baseline=(v.base)]
    \coordinate (v) at (0,.1);
    \coordinate (v0) at (0,0);
    \path (v0) + (90:1) coordinate (v1);
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    \path (v0) + (378:1) coordinate (v5);
    \draw (v2) -- (v1);
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    \draw (v4) -- (v0);
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    \node[circ] at (v3) {1};
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    \node[circ] at (v0) {4};
    \node[circ] at (v1) {5};
    \node at (0,-1.4) {$S_{40}$};
    \end{tikzpicture}
    \begin{tikzpicture}[very thick, baseline=(v.base)]
    \coordinate (v) at (0,.1);
    \coordinate (v0) at (0,0);
    \path (v0) + (90:1) coordinate (v1);
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    \draw (v2) -- (v1);
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    \node[circ] at (v4) {4};
    \node[circ] at (v1) {5};
    \node at (0,-1.4) {$S_{41}$};
    \end{tikzpicture}
    \begin{tikzpicture}[very thick, baseline=(v.base)]
    \coordinate (v) at (0,.1);
    \coordinate (v0) at (0,0);
    \path (v0) + (90:1) coordinate (v1);
    \path (v0) + (162:1) coordinate (v2);
    \path (v0) + (234:1) coordinate (v3);
    \path (v0) + (306:1) coordinate (v4);
    \path (v0) + (378:1) coordinate (v5);
    \draw (v3) -- (v0);
    \draw (v4) -- (v5);
    \draw (v2) -- (v1);
    \draw (v2) -- (v0);
    \draw (v5) -- (v1);
    \draw (v5) -- (v0);
    \draw (v1) -- (v0);
    \draw[dashed] (v3) -- (v4);
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    \node[circ] at (v4) {1};
    \node[circ] at (v2) {2};
    \node[circ] at (v5) {3};
    \node[circ] at (v1) {4};
    \node[circ] at (v0) {5};
    \node at (0,-1.4) {$S_{42}$};
    \end{tikzpicture}
    \begin{tikzpicture}[very thick, baseline=(v.base)]
    \coordinate (v) at (0,.1);
    \coordinate (v0) at (0,0);
    \path (v0) + (90:1) coordinate (v1);
    \path (v0) + (162:1) coordinate (v2);
    \path (v0) + (234:1) coordinate (v3);
    \path (v0) + (306:1) coordinate (v4);
    \path (v0) + (378:1) coordinate (v5);
    \draw (v1) -- (v2);
    \draw (v1) -- (v5);
    \draw (v3) -- (v2);
    \draw (v4) -- (v0);
    \draw (v4) -- (v5);
    \draw (v0) -- (v2);
    \draw (v0) -- (v5);
    \draw (v2) -- (v5);
    \draw[dashed] (v3) -- (v4);
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    \node[circ] at (v3) {1};
    \node[circ] at (v4) {2};
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    \node[circ] at (v2) {4};
    \node[circ] at (v5) {5};
    \node at (0,-1.4) {$S_{43}$};
    \end{tikzpicture}
    \begin{tikzpicture}[very thick, baseline=(v.base)]
    \coordinate (v) at (0,.1);
    \coordinate (v0) at (0,0);
    \path (v0) + (90:1) coordinate (v1);
    \path (v0) + (162:1) coordinate (v2);
    \path (v0) + (234:1) coordinate (v3);
    \path (v0) + (306:1) coordinate (v4);
    \path (v0) + (378:1) coordinate (v5);
    \draw (v1) -- (v2);
    \draw (v3) -- (v2);
    \draw (v0) -- (v5);
    \draw (v0) -- (v4);
    \draw (v0) -- (v2);
    \draw (v5) -- (v4);
    \draw (v5) -- (v2);
    \draw[dashed] (v3) -- (v4);
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    \node[circ] at (v3) {1};
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    \node[circ] at (v4) {4};
    \node[circ] at (v2) {5};
    \node at (0,-1.4) {$S_{44}$};
    \end{tikzpicture}
    \begin{tikzpicture}[very thick, baseline=(v.base)]
    \coordinate (v) at (0,.1);
    \coordinate (v0) at (0,0);
    \path (v0) + (90:1) coordinate (v1);
    \path (v0) + (162:1) coordinate (v2);
    \path (v0) + (234:1) coordinate (v3);
    \path (v0) + (306:1) coordinate (v4);
    \path (v0) + (378:1) coordinate (v5);
    \draw (v3) -- (v2);
    \draw (v4) -- (v5);
    \draw (v1) -- (v0);
    \draw (v1) -- (v5);
    \draw (v1) -- (v2);
    \draw (v0) -- (v5);
    \draw (v0) -- (v2);
    \draw[dashed] (v3) -- (v4);
    \node[circ] at (v3) {0};
    \node[circ] at (v4) {1};
    \node[circ] at (v1) {2};
    \node[circ] at (v0) {3};
    \node[circ] at (v5) {4};
    \node[circ] at (v2) {5};
    \node at (0,-1.4) {$S_{45}$};
    \end{tikzpicture}
    \begin{tikzpicture}[very thick, baseline=(v.base)]
    \coordinate (v) at (0,0);
    \coordinate (v0) at (.9,.9);
    \coordinate (v1) at (-.9,.9);
    \coordinate (v2) at (-.9,-.9);
    \coordinate (v3) at (.9,-.9);
    \coordinate (v4) at (.4,0);
    \coordinate (v5) at (-.4,0);
    \draw (v2) -- (v1);
    \draw (v5) -- (v4);
    \draw (v5) -- (v0);
    \draw (v5) -- (v1);
    \draw (v3) -- (v4);
    \draw (v3) -- (v0);
    \draw (v4) -- (v0);
    \draw (v4) -- (v1);
    \draw (v0) -- (v1);
    \draw[dashed] (v2) -- (v3);
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    \node[circ] at (v5) {1};
    \node[circ] at (v3) {2};
    \node[circ] at (v4) {3};
    \node[circ] at (v0) {4};
    \node[circ] at (v1) {5};
    \node at (0,-1.5) {$S_{46}$};
    \end{tikzpicture}
    \begin{tikzpicture}[very thick, baseline=(v.base)]
    \coordinate (v) at (0,0);
    \coordinate (v0) at (.9,.9);
    \coordinate (v1) at (-.9,.9);
    \coordinate (v2) at (-.9,-.9);
    \coordinate (v3) at (.9,-.9);
    \coordinate (v4) at (.4,0);
    \coordinate (v5) at (-.4,0);
    \draw (v2) -- (v1);
    \draw (v2) -- (v5);
    \draw (v3) -- (v0);
    \draw (v3) -- (v4);
    \draw (v0) -- (v4);
    \draw (v0) -- (v1);
    \draw (v0) -- (v5);
    \draw (v4) -- (v1);
    \draw (v4) -- (v5);
    \draw (v1) -- (v5);
    \draw[dashed] (v2) -- (v3);
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    \node[circ] at (v3) {1};
    \node[circ] at (v0) {2};
    \node[circ] at (v4) {3};
    \node[circ] at (v1) {4};
    \node[circ] at (v5) {5};
    \node at (0,-1.5) {$S_{47}$};
    \end{tikzpicture}
    \begin{tikzpicture}[very thick, baseline=(v.base)]
    \coordinate (v) at (0,0);
    \coordinate (v0) at (.9,.9);
    \coordinate (v1) at (-.9,.9);
    \coordinate (v2) at (-.9,-.9);
    \coordinate (v3) at (.9,-.9);
    \coordinate (v4) at (.4,0);
    \coordinate (v5) at (-.4,0);
    \draw (v2) -- (v5);
    \draw (v2) -- (v1);
    \draw (v3) -- (v0);
    \draw (v3) -- (v4);
    \draw (v0) -- (v4);
    \draw (v0) -- (v1);
    \draw (v5) -- (v4);
    \draw (v5) -- (v1);
    \draw (v4) -- (v1);
    \draw[dashed] (v2) -- (v3);
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    \node[circ] at (v3) {1};
    \node[circ] at (v0) {2};
    \node[circ] at (v5) {3};
    \node[circ] at (v4) {4};
    \node[circ] at (v1) {5};
    \node at (0,-1.5) {$S_{48}$};
    \end{tikzpicture}
    \begin{tikzpicture}[very thick, baseline=(v.base)]
    \coordinate (v) at (0,0);
    \coordinate (v0) at (.9,.9);
    \coordinate (v1) at (-.9,.9);
    \coordinate (v2) at (-.9,-.9);
    \coordinate (v3) at (.9,-.9);
    \coordinate (v4) at (.4,0);
    \coordinate (v5) at (-.4,0);
    \draw (v4) -- (v3);
    \draw (v4) -- (v0);
    \draw (v2) -- (v1);
    \draw (v2) -- (v5);
    \draw (v1) -- (v5);
    \draw (v1) -- (v0);
    \draw (v5) -- (v0);
    \draw (v3) -- (v0);
    \draw[dashed] (v2) -- (v3);
    \node[circ] at (v4) {0};
    \node[circ] at (v2) {1};
    \node[circ] at (v1) {2};
    \node[circ] at (v5) {3};
    \node[circ] at (v3) {4};
    \node[circ] at (v0) {5};
    \node at (0,-1.5) {$S_{49}$};
    \end{tikzpicture}
    \caption{Additional minimal forbidden subgraphs for $\mathcal{D}_\infty^\pm$ (up to switching equivalence).} \label{fig:minimal-forb-2}
\end{figure}

Next, generalizing \cref{lem:computation-1}, we carry out the following computation.

\begin{lemma} \label{lem:computation-2}
    For every minimal forbidden subgraph $F^\pm$ for the family $\mathcal{D}^\pm_\infty$, and every nonempty signed vertex subset $A^\pm$ of $F^\pm$, the path extensions and the clique extensions of $F^\pm$ satisfy
    \[
        \lim_{\ell \to \infty} \lambda_1(F^\pm, A^\pm, \ell) \le -\lambda^* \quad \text{and} \quad \lim_{m \to \infty} \lambda_1(F^\pm, A^\pm, K_m) \le -\lambda^*.
    \]
\end{lemma}

In \cref{sec:numerics}, we prove, under computer assistance, that the inequalities are strict in \cref{lem:computation-2} except when $F^\pm = G_{23}$ and $A^\pm \in \sset{\sset{1^+}, \sset{1^-}, \sset{2^+}, \sset{2^-}}$.

Lastly, the proof of \cref{lem:path-clique-extension} can be taken almost verbatim to show the following generalization.

\begin{lemma} \label{lem:path-clique-extension-signed}
    Suppose that $A^\pm$ is a nonempty vertex subset of a signed graph $F^\pm$ and $\lambda \ge 2$. If the path extensions of $F^\pm$ satisfy
    \[
        \lim_{\ell \to \infty} \lambda_1(F^\pm, A^\pm, \ell) < -\lambda,
    \]
    then there exists $m \in \mathbb{N}^+$ such that the path-clique extensions of $F^\pm$ satisfy
    \[
        \pushQED{\qed}
        \lambda_1(F^\pm, A^\pm, \ell, K_m) < -\lambda \text{ for every }\ell \in \mathbb{N}.\qedhere
        \popQED
    \]
\end{lemma}

We are ready to prove \cref{thm:main2-2}.

\begin{proof}[Proof of \cref{thm:main2-2}]
    Suppose that $\lambda \in [2, \lambda^*)$. Denote $\mathcal{F}^\pm$ the set of minimal forbidden subgraphs for the family $\mathcal{D}^\pm$. Combining \cref{lem:computation-2,lem:path-clique-extension-signed}, we choose $\ell, m \in \mathbb{N}^+$ such that for every $F^\pm \in \mathcal{F}^\pm$, the extension family $\mathcal{X}^\pm(F^\pm, \ell, m)$ is disjoint from $\G^\pm(\la)$. We know that $S_5 \not\in \G^\pm(\la)$ and $-K_4 \not\in \G^\pm(\la)$. In particular, no graph in $\G^\pm(\la)$ contains a subgraph that is switching equivalent to any member in the following family
    \begin{equation} \label{eqn:main2-2}
        \sset{S_5, -K_4} \cup \bigcup_{F^\pm \in \mathcal{F}^\pm} \mathcal{X}^\pm(F^\pm, \ell, m).
    \end{equation}
    Using \cref{lem:forbid-extensions-signed}, we obtain $N \in \mathbb{N}$ such that for every connected signed graph $G^\pm$ with more than $N$ vertices, if no member in \eqref{eqn:main2-2} is switching equivalent to a subgraph of $G^\pm$, then neither is any $F^\pm \in \mathcal{F}^\pm$, and so $G^\pm$ is represented by a subset of $D_n$ and is clearly in $\mathcal{G}^\pm(2)$. This implies that every connected graph in $\G^\pm(\la) \setminus \mathcal{G}^\pm(2)$ has at most $N$ vertices.
\end{proof}

\subsection{Proof of \cref{thm:main2} for $\lambda \ge \lambda^*$}

\begin{proof}[Proof of \cref{thm:main2} for $\lambda \ge \lambda^*$]
    If $\mathcal{F}^\pm$ is a finite forbidden subgraph characterization for $\G^\pm(\la)$, then the all-positive signed graphs in $\mathcal{F}^\pm$ would form a finite forbidden subgraph characterization for $\G(\la)$, which contradicts with \cref{thm:main1} for $\lambda \ge \lambda^*$.
\end{proof}

\section{Applications} \label{sec:app}

\subsection{Symmetric integer matrices}

As was observed by Vijayakumar in \cite[Theorem~4.2]{V87}, among all symmetric integer matrices whose diagonal entries are all zero, the signed adjacency matrices of signed graphs are most difficult to test whether their smallest eigenvalues are less than $-2$. This observation leads to a connection between \cref{thm:main2} and \cref{cor:mat}.

\begin{proof}[Proof of \cref{cor:mat} for $\lambda < \lambda^*$]
    Suppose $\lambda < \lambda^*$. Let $\mathcal{F}^\pm$ be a finite forbidden subgraph characterization for $\G^\pm(\la)$, and let $n$ be the maximum number of vertices in a signed graph in $\mathcal{F}^\pm$. Let $A = (a_{ij})$ be a symmetric integer matrix such that $\lambda_1(A) < -\lambda$ and $a_{ii} = 0$ for every $i$.

    Suppose for a moment that $\abs{a_{ij}} > \lambda$ for some $i < j$. Then the eigenvalues of the principal submatrix
    \[
        \begin{pmatrix}
            a_{ii} & a_{ij} \\
            a_{ji} & a_{jj}
        \end{pmatrix}
    \]
    are $\pm a_{ij}$, one of which is less than $-\lambda$. Hereafter we may assume that every entry of $A$ is at most $\lambda$ in absolute value. We break the rest of the proof into two cases.

    \paragraph{Case 1:} $\abs{a_{ij}} = 2$ for some $i < j$. In this case, $2 \le \lambda < \lambda^*$, and so $\lambda_1(A) < -2$. By taking a proper similarity transformation, we may assume without loss of generality that $a_{12} = 2$. One can verify by brute force that the smallest eigenvalue of
    \[
        \begin{pmatrix}
            0 & 2 & a \\
            2 & 0 & a' \\
            a & a' & 0
        \end{pmatrix}
    \]
    is at most $-\sqrt5$ for all $a, a' \in \sset{0, \pm 1, \pm 2}$ with $a \neq a'$. We may assume that $a_{1k} = a_{2k}$ for every $k \ge 3$ because otherwise we would have found a principal submatrix of order $3$ that has an eigenvalue less than $-\lambda$. Observe that
    \[
        \bm{y} = \begin{pmatrix}
            1 & -1 & 0 & 0 & \cdots & 0 & 0
        \end{pmatrix}^\intercal
    \]
    is an eigenvector of $A$ associated with $-2$. Therefore an eigenvector $\bm{x}$ of $A$ associated with $\lambda_1(A)$, which is less than $-2$, satisfies $x_1 = x_2$ because $\bm{x}$ is orthogonal to $\bm{y}$. Let $A_1$ be the principal submatrix obtained by removing the first row and the first column of $A$, and let $A_1'$ be the matrix obtained by changing the first diagonal entry from $0$ to $1$. One can check that
    \[
        \begin{pmatrix}
            x_1 + x_2 & x_3 & x_4 & \cdots
        \end{pmatrix}^\intercal
    \]
    is an eigenvector of $A_1'$ associated with $\lambda_1(A)$, and so $\lambda_1(A_1') \le \lambda_1(A)$. Since $A_1' - A_1$ is positive semidefinite, we have $\lambda_1(A_1) \le \lambda_1(A_1') \le \lambda_1(A) < -\lambda$. We have successfully found a proper principal submatrix $A_1$ of $A$ with $\lambda_1(A_1) < -\lambda$, and we can recurse on $A_1$.

    \paragraph{Case 2:} $\abs{a_{ij}} \le 1$ for every $i < j$. Then $A$ is the adjacency matrix of a signed graph, say $G^\pm$. Since $G^\pm \not\in \G^\pm(\la)$, the signed graph $G^\pm$ contains some $F^\pm \in \mathcal{F}^\pm$ as a subgraph. Clearly $A_{F^\pm}$ is a principal submatrix of $A$ such that $A_{F^\pm}$ is of order at most $n$, and its smallest eigenvalue is less than $-\lambda$.
\end{proof}

\begin{proof}[Proof of \cref{cor:mat} for $\lambda \ge \lambda^*$]
    Since $\G^\pm(\la)$ does not have a finite forbidden subgraph characterization, there exists a sequence of minimal forbidden subgraphs $F^\pm_1, F^\pm_2, \dots$ for $\G^\pm(\la)$ with increasing number of vertices. Let $A_n$ be the signed adjacency matrix of $F^\pm_n$. By the choice of $F^\pm_n$, we know that $\lambda_1(A_n) < -\lambda$, and moreover no proper principal submatrix of $A_n$ has an eigenvalue less than $-\lambda$.
\end{proof}

\subsection{Spherical two-distance sets} \label{subsec:two-distance}

We introduce the definition of chromatic number for signed graphs.

\begin{definition}[Chromatic number] \label{def:chromatic}
    A \emph{valid $p$-coloring} of a signed graph $G^\pm$ is a coloring of the vertices using $p$ colors such that the endpoints of every negative edge receive different colors, and the endpoints of every positive edge receive identical colors. (See \cref{fig:coloring} for an example.) The \emph{chromatic number} $\chi(G^\pm)$ of a signed graph $G^\pm$ is the smallest $p$ for which $G^\pm$ has a valid $p$-coloring. If $G^\pm$ does not have a valid $p$-coloring for any $p$, we write $\chi(G^\pm) = \infty$.
\end{definition}

\begin{figure}[t]
    \centering
    \begin{tikzpicture}[very thick,scale=.8]
        \coordinate (a) at (0.922649731,3);
        \coordinate (b) at (2.077350269,3);
        \coordinate (c) at (-0.577350269,1);
        \coordinate (d) at (0.577350269,1);
        \coordinate (e) at (0,0);
        \coordinate (f) at (2.422649731,1);
        \coordinate (g) at (3.577350269,1);
        \coordinate (h) at (3,0);
        \draw [fill=red!30,draw=none] (0,0.65) circle[radius=1];
        \draw [fill=blue!30,draw=none] (3,0.65) circle[radius=1];
        \draw [fill=green!30,draw=none] (1.5,3) circle[radius=1];
        \draw[dashed] (c) -- (a) -- (d);
        \draw[dashed] (g) -- (b) -- (f);
        \draw[dashed] (d) -- (h);
        \draw[dashed] (e) -- (f);
        \draw (a) node[vertex]{} -- (b) node[vertex]{};
        \draw (c) node[vertex]{} -- (e) node[vertex]{} -- (d) node[vertex]{};
        \draw (f) node[vertex]{} -- (g) node[vertex]{} -- (h) node[vertex]{} -- cycle;
    \end{tikzpicture}
    \caption{A valid $3$-coloring of a signed graph.} \label{fig:coloring}
\end{figure}

Recall from \cref{sec:intro} the following spectral graph quantity,
\begin{equation} \label{def:kpl}
    k_p(\la) = \inf\dset{\frac{\abs{G^\pm}}{\operatorname{mult}(\lambda, G^\pm)}}{\chi(G^\pm) \le p \text{ and }\lambda^1(G^\pm) = \lambda}.
\end{equation}
We say that $k_p(\lambda)$ is \emph{achievable} if it is finite and the infimum can be attained. The spectral graph quantity $k_p(\lambda)$ can be seen as a generalization of the \emph{spectral radius order} $k(\lambda)$ defined by
\[
    k(\lambda) = \min\dset{\abs{G}}{\lambda^1(G) = \lambda}.
\]
Indeed, $k_1(\lambda) = k_2(\lambda) = k(\lambda)$, however, the behavior of $k_p(\lambda)$ is far more mysterious when $p \ge 3$ (see the remark after \cite[Definition~1.5]{JTYZZ20} for more details). The technique developed in the current paper enables us to certify values of $k_p(\la)$ whenever $\lambda < \lambda^*$.

\begin{proposition} \label{thm:kpl}
    For every $p \ge 3$ and $\lambda < \lambda^*$, there exists $N \in \mathbb{N}$ such that
    \[
        k_p(\la) = \min\dset{\frac{\abs{G^\pm}}{\operatorname{mult}(\lambda, G^\pm)}}{\abs{G^\pm} \le N, \chi(G^\pm) \le p \text{ and }\lambda^1(G^\pm) = \lambda},
    \]
    and in particular, $k_p(\lambda)$ is achievable whenever it is finite.
\end{proposition}

We will return to the proof of \cref{thm:kpl} towards the end of the subsection. The next result constructs large spherical codes with two fixed angles.

\begin{proposition}[Proposition~1.6 of Jiang et al.~\cite{JTYZZ20}] \label{thm:lower-bound}
    Fix $-1 \le \beta < 0 \le \alpha < 1$. Then $N_{\al,\be}(d) \ge d$ for every $d \in \mathbb{N}^+$. Moreover if $k_p(\la) < \infty$, where $\lambda =(1-\alpha)/(\alpha - \beta)$ and $p = \lfloor -\alpha/\beta \rfloor + 1$, then
    \begin{equation*}
        \pushQED{\qed}
        N_{\alpha,\beta}(d) \ge \begin{dcases}
            \frac{k_p(\la) d}{k_p(\la)-1} - O_{\alpha,\beta}(1) & \text{if }k_p(\la) \text{ is achievable}, \\
            \frac{k_p(\la) d}{k_p(\la)-1} - o(d) & \text{otherwise}.
        \end{dcases} \qedhere
        \popQED
    \end{equation*}
\end{proposition}

\cref{conj:main} asserts that the constructions in \cref{thm:lower-bound} are optimal up to an error sublinear in $d$. A framework to bound $N_{\al,\be}(d)$ from above was formulated in \cite{JTYZZ20}.

\begin{definition}[Definition~1.10 of Jiang et al.~\cite{JTYZZ20}] \label{def:mph}
    Given $p \in \mathbb{N}^+$ and a family $\mathcal{H}$ of signed graphs, let $M_{p, \mathcal{H}}(\lambda, N)$ be the maximum possible value of $\operatorname{mult}(\lambda, G^\pm)$ over all signed graphs $G^\pm$ on at most $N$ vertices that do not contain any member of $\mathcal{H}$ as a subgraph and satisfy $\chi(G^\pm) \le p$ and $\lambda^{p+1}(G^\pm) \le \lambda$.
\end{definition}

\begin{theorem}[Theorem~1.11 of Jiang et al.~\cite{JTYZZ20}] \label{thm:upper-bound}
    Fix $-1 \le \beta < 0 \le \alpha < 1$. Set $\lambda = (1-\alpha)/(\alpha - \beta)$ and $p = \lfloor -\alpha/\beta \rfloor + 1$. Let $\mathcal{H}$ be a finite family of signed graphs with $\lambda^1(H^\pm) > \lambda$ for each $H^\pm \in \mathcal{H}$. Then
    \begin{equation*}
        \pushQED{\qed}
        N_{\alpha,\beta}(d) \le d + M_{p, \mathcal{H}}(\lambda, N_{\al,\be}(d)) + O_{\alpha, \beta}(1).  \qedhere
        \popQED
    \end{equation*}
\end{theorem}

A proper choice of the finite family $\mathcal{H}$ in the last theorem establishes \cref{conj:main} for $\lambda < \lambda^*$.

\begin{corollary} \label{cor:spherical}
    Fix $-1 \le \beta < 0 \le \alpha < 1$. Set $\lambda = (1-\alpha)/(\alpha-\beta)$ and $p = \lfloor -\alpha / \beta \rfloor + 1$. If $\lambda < \lambda^*$, then
    \[
        N_{\al,\be}(d) = \begin{dcases}
            \frac{k_p(\la) d}{k_p(\la)-1} + O_{\alpha,\beta}(1) & \text{if }k_p(\la) < \infty, \\
            d + O_{\alpha,\beta}(1) & \text{otherwise}.
        \end{dcases}
    \]
\end{corollary}

\begin{proof}
    According to \cref{cor:main2}, we can take a finite forbidden subgraph characterization, denoted $\mathcal{H}$, for the family $\G^\mp(\la)$ of signed graphs with largest eigenvalue at most $\lambda$. Thus $M_{p, \mathcal{H}}(\lambda, N)$ is the maximum possible value of $\operatorname{mult}(\lambda, G^\pm)$ over all signed graphs $G^\pm$ on at most $N$ vertices that satisfy $\chi(G^\pm) \le p$ and $\lambda^1(G^\pm) \le \lambda$. Note that $M_{p, \mathcal{H}}(\lambda, N)$ is a non-decreasing function of $N$. We break the rest of the proof into two cases.

    \paragraph{Case 1:} $M_{p, \mathcal{H}}(\lambda, N) = 0$ for every $N \in \mathbb{N}^+$. In other words, there is no signed graph $G^\pm$ that satisfy $\chi(G^\pm) \le p$ and $\lambda^1(G^\pm) = \lambda$. Thus this case corresponds to the case where $k_p(\la) = \infty$. According to \cref{thm:lower-bound,thm:upper-bound}, we have $N_{\al,\be}(d) = d + O_{\alpha,\beta}(1)$.

    \paragraph{Case 2:} $M_{p, \mathcal{H}}(\lambda, N) > 0$ for every $N \ge d_0$. This case corresponds to the case where $k_p(\la) < \infty$. Suppose that $d \ge d_0$. \cref{thm:lower-bound} says that $N_{\al,\be}(d) \ge d$, and so $M_{p, \mathcal{H}}(\lambda, N_{\al,\be}(d)) > 0$. Let $G^\pm$ be a signed graph with at most $N_{\alpha,\beta}(d)$ vertices that satisfy $\chi(G^\pm) \le p$, $\lambda^1(G^\pm) \le \lambda$ and $\operatorname{mult}(\lambda, G^\pm) = M_{p, \mathcal{H}}(\lambda, N_{\al,\be}(d))$. Thus
    \[
        k_p(\la) \le \frac{\abs{G^\pm}}{\operatorname{mult}(\lambda, G^\pm)} \le \frac{N_{\al,\be}(d)}{M_{p, \mathcal{H}}(\lambda, N_{\al,\be}(d))}.
    \]
    \cref{thm:upper-bound} then implies that
    \[
        N_{\al,\be}(d) \le \frac{k_p(\la) d}{k_p(\la)-1} + O_{\alpha,\beta}(1),
    \]
    which, in view of \cref{thm:kpl,thm:lower-bound}, gives $N_{\al,\be}(d) = k_p(\la) d / (k_p(\la) - 1) + O_{\alpha,\beta}(1)$.
\end{proof}

Lastly, we come back to the achievability of spectral graph quantity $k_p(\la)$. Actually we may assume in the definition \eqref{def:kpl} of $k_p(\la)$ that $G^\pm$ is connected by passing to a suitable connected component of $G^\pm$ in case $G^\pm$ is not connected. Furthermore, as we shall mainly work with the signed graph that negates all the edge signs of $G^\pm$ in \eqref{def:kpl}, we redefine $k_p(\la)$ as follows:
\[
    k_p(\la) = \inf\dset{\frac{\abs{G^\pm}}{\operatorname{mult}(-\lambda, G^\pm)}}{G^\pm \text{ is connected}, \chi(-G^\pm) \le p, \text{ and }\lambda^1(G^\pm) = -\lambda},
\]
where $-G^\pm$ negates all the edge signs of $G^\pm$.

We break the proof of \cref{thm:kpl} into three cases $\lambda < 2$, $\lambda \in (2, \lambda^*)$ and $\lambda = 2$.

\begin{proof}[Proof of \cref{thm:kpl} for $\lambda < 2$]
    We obtain $\ell \in \mathbb{N}^+$ from \cref{lem:computation-1}(\ref{lem:p}) such that $P_{\ell+1} \not\in \G^\pm(\la)$. Applying \cref{lem:forbid-extensions-signed} to $F^\pm = K_1$ (the $1$-vertex graph), $k_1 = 4$, $k_2 = 3$ and $m = 2p$, we obtain $N \in \mathbb{N}$ such that for every connected signed graph $G^\pm$, if $G^\pm$ does not contain any subgraph that is switching equivalent to any member in $\sset{S_4, -K_3} \cup \mathcal{X}^\pm(K_1, \ell, 2p)$, then $G^\pm$ contains at most $N$ vertices.

    It suffices to show that for every connected signed graph $G^\pm$ with more than $N$ vertices, either $\chi(-G^\pm) > p$ or $G^\pm \not\in \G^\pm(\la)$. By our choice of $N$, $G^\pm$ contains a subgraph that is switching equivalent to a member in $\sset{S_4, -K_3} \cup \mathcal{X}^\pm(K_1, \ell, 2p)$. We break the rest of the proof into two cases.

    \paragraph{Case 1:} $G^\pm$ contains a subgraph that is switching equivalent to $S_4$, $-K_3$, or a path extension in $\mathcal{X}^\pm(K_1, \ell, 2p)$. Since $\lambda_1(S_4) = \lambda_1(-K_3) = -2$ and every path extension in $\mathcal{X}^\pm(K_1, \ell, 2p)$ is switching equivalent to $P_{\ell+1}$, the signed graph $G^\pm$ is not in $\G^\pm(\la)$.

    \paragraph{Case 2:} $G^\pm$ contains a subgraph that is switching equivalent to a path-clique extension or a clique extension in $\mathcal{X}^\pm(K_1, \ell, 2p)$. Observe that every path-clique extension and every clique extension in $\mathcal{X}^\pm(K_1, \ell, 2p)$ contains a subgraph that is switching equivalent to $K_{2p+1}$, and moreover every signed graph that is switching equivalent to $K_{2p+1}$ always contains $K_{p+1}$ as a subgraph. Thus the signed graph $G^\pm$ contains $K_{p+1}$ as a subgraph. Therefore $-G^\pm$ contains $-K_{p+1}$ as a subgraph, and so $\chi(-G^\pm) \ge p+1$.
\end{proof}

The proof of \cref{thm:kpl} for $\lambda \in (2, \lambda^*)$ follows immediately from \cref{thm:main2-2}.

\begin{proof}[Proof of \cref{thm:kpl} for $\lambda \in (2, \lambda^*)$]
    \cref{thm:main2-2} implies that there exists $N\in\mathbb{N}$ such that every connected signed graph $G^\pm$ with $\lambda_1(G^\pm) = -\lambda$ has at most $N$ vertices.
\end{proof}

The proof of \cref{thm:kpl} for $\lambda = 2$ is trickier. We need the following generalization of \cref{thm:cgss}. This generalization was explicitly recognized by Chawathe and Vijayakumar~\cite[Theorem~1.1]{CV90}, who attributed the result to Witt~\cite{W41}, and referred the reader to \cite{CGSS76} for a combinatorial proof.

\begin{theorem}[Witt~\cite{W41} and Cameron et al.~\cite{CGSS76}] \label{thm:cgss-signed}
    For every connected signed graph $G^\pm$, the smallest eigenvalue of $G^\pm$ is at least $-2$ if and only if $G^\pm$ is represented by a subset of $D_n$ or $E_8$.
\end{theorem}

Recall that $\mathcal{D}_\infty^\pm$ denotes the family of signed graphs that are represented by a subset of $D_n$, where
\[
    D_n := \dset{\varepsilon_1 \bm{e}_i + \varepsilon_2 \bm{e}_j}{\varepsilon_1, \varepsilon_2 = \pm 1, 1 \le i < j \le n}.
\]
We shall focus on the signed graphs in $\mathcal{D}_\infty^\pm$. To that end, we generalize bidirected graphs to multigraphs as follows.

\begin{definition}[Bidirected multigraph]
    A \emph{bidirected multigraph} is a multigraph (allowing parallel edges, but no loops) in which each vertex-edge incidence $(v, e)$ has a positive or negative sign such that every pair of parallel edges $e_1$ and $e_2$ satisfies the following condition: for one of the shared endpoints, say $v_1$, the incidences $(v_1, e_1)$ and $(v_1, e_2)$ have the same sign, while for the other shared endpoint $v_2$, the incidences $(v_2, e_1)$ and $(v_2, e_2)$ have opposite signs. In particular, there are at most two edges between any two vertices in a bidirected multigraph.
\end{definition}

As we transfer relevant properties of a signed graph $G^\pm \in \mathcal{D}_\infty^\pm$ to a bidirected multigraph $\vec{H}$, a vertex coloring of $G^\pm$ will correspond to an edge coloring of $\vec{H}$.

\begin{definition}[Intersecting edges and proper edge coloring] \label{def:proper-edge-coloring}
    As opposed to parallel edges, we say that two edges of a bidirected multigraph \emph{intersect} at a vertex $v$ if $v$ is the only vertex they have in common. A \emph{proper $p$-edge-coloring} of a bidirected multigraph $\vec{H}$ is a coloring of the edges using $p$ colors such that for each pair of edges $e_1, e_2$ that intersect at one vertex, say $v$, the edges $e_1$ and $e_2$ receive different colors when the incidences $(v, e_1)$ and $(v, e_2)$ have the same sign, whereas $e_1$ and $e_2$ receive identical colors when $(v, e_1)$ and $(v, e_2)$ have opposite signs. See \cref{fig:bidirected-multigraph} for an example.
\end{definition}

\begin{figure}[t]
    \centering
    \begin{tikzpicture}[
        very thick, baseline=(v.base),
        neg/.pic = {\draw[fill=white] (0,0) circle (0.1); \draw (-0.1,0)--(0.1,0);},
        pos/.pic = {\draw[fill=white] (0,0) circle (0.1); \draw (-0.1,0)--(0.1,0); \draw (0,0.1)--(0,-0.1);},
    ]
        \node[vertex] (a) at (-2,0) {};
        \node[vertex] (b) at (0,0) {};
        \node[vertex] (c) at (2,0) {};
        \node[vertex] (d) at (4,0) {};
        \node[vertex] (e) at (6,0) {};
        \node[vertex] (f) at (8,0) {};
        \path[color=blue] (a) edge[bend left=45] (b);
        \path[color=blue] (a) edge[bend left=-45] (b);
        \path[color=green] (b) edge[bend left=45] (c);
        \path[color=red] (b) edge[bend left=-45] (c);
        \draw[color=green] (c) -- (d);
        \path[color=green] (d) edge[bend left=45] (e);
        \path[color=red] (d) edge[bend left=-45] (e);
        \path[color=blue] (e) edge[bend left=45] (f);
        \path[color=blue] (e) edge[bend left=-45] (f);
        \draw (-1.6,0.3) pic {pos};
        \draw (-1.6,-0.3) pic {neg};
        \draw (-0.4,0.3) pic {pos};
        \draw (0.4,0.3) pic {pos};
        \draw (-0.4,-0.3) pic {pos};
        \draw (0.4,-0.3) pic {pos};
        \draw (1.6,0.3) pic {pos};
        \draw (1.6,-0.3) pic {neg};
        \draw (2.5,0) pic {neg};
        \draw (3.5,0) pic {pos};
        \draw (4.4,0.3) pic {neg};
        \draw (4.4,-0.3) pic {pos};
        \draw (5.6,0.3) pic {neg};
        \draw (5.6,-0.3) pic {neg};
        \draw (6.4,0.3) pic {neg};
        \draw (6.4,-0.3) pic {neg};
        \draw (7.6,0.3) pic {neg};
        \draw (7.6,-0.3) pic {pos};
    \end{tikzpicture}
    \caption{A proper $3$-edge coloring of a bidirected multigraph.} \label{fig:bidirected-multigraph}
\end{figure}

Now we are ready to prove the last case of \cref{thm:kpl}.

\begin{proof}[Proof of \cref{thm:kpl} for $\lambda = 2$]
    Let $G^\pm$ be an arbitrary connected signed graph with $\chi(-G^\pm) \le p$ and $\lambda_1(G^\pm) = 2$. \cref{thm:cgss-signed} says that $G^\pm$ is represented by a subset of $D_n$ or $E_8$. Because there are only finitely many signed graphs that are represented by a subset of $E_8$, it suffices to show that
    \begin{gather}
        \frac{\abs{G^\pm}}{\operatorname{mult}(-2, G^\pm)} = \left(\frac{p}{p-1}\right)^2 \text{ for some }G^\pm \in \mathcal{D}_\infty^\pm, \text{ and} \label{eqn:kpl-1}\\
        \frac{\abs{G^\pm}}{\operatorname{mult}(-2, G^\pm)} \ge \left(\frac{p}{p-1}\right)^2 \text{ for every }G^\pm \in \mathcal{D}_\infty^\pm \text{ with }\abs{G^\pm} \ge p^2. \label{eqn:kpl-2}
    \end{gather}

    Suppose that $G^\pm$ is represented by $V \subseteq D_n$. Construct a bidirected multigraph $\vec{H}$ with vertices $v_1, \dots, v_n$ and incidence signing $\sigma$ as follows. For each vector in $V$ of the form $\varepsilon_1 \bm{e}_i + \varepsilon_2 \bm{e}_j$ with $i < j$, we put an edge $e$ connecting $v_i$ and $v_j$, and we assign $\sigma(v_i, e) = \varepsilon_1$ and $\sigma(v_j, e) = \varepsilon_2$. For every pair of parallel edges $e_1$ and $e_2$, which connect $v_i$ and $v_j$, since $\sigma(v_i, e_1)\bm{e}_i + \sigma(v_j, e_1)\bm{e}_j$ and $\sigma(v_i, e_2)\bm{e}_i + \sigma(v_j, e_2)\bm{e}_j$ are two distinct vectors from $V$, their inner product satisfies
    \[
        \sset{0, \pm2} \ni \sigma(v_i, e_1)\sigma(v_i, e_2) + \sigma(v_j, e_1)\sigma(v_j, e_2) = A_{G^\pm}(e_1,e_2) \in \sset{0,\pm 1},
    \]
    and so their inner product must be $0$. Therefore $\vec{H}$ is a bona fide bidirected multigraph. We may remove all the isolated vertices (if any) from $\vec{H}$.

    Conversely one can read off the vectors in $V$ from $\vec{H}$ --- for every edge $e$, which connects $v_i$ and $v_j$ with $i < j$, in $\vec{H}$, set
    \[
        \vec{v}(e) := \sigma(v_i, e) \bm{e}_i + \sigma(v_j, e) \bm{e}_j \in D_n.
    \]
    Furthermore one can reconstruct $G^\pm$ from $\vec{H}$ as follows. We identify the vertex set of $G^\pm$ with the edge set of $\vec{H}$, and for each pair of distinct vertices $e_1$ and $e_2$ of $G^\pm$, they are adjacent in $G^\pm$ if and only if $e_1$ and $e_2$ intersect at one vertex, say $v$, in $\vec{H}$, and the sign of the edge $e_1e_2$ in $G^\pm$ is $\sigma(v, e_1)\sigma(v, e_2)$. One can then check that the reconstructed $G^\pm$ is indeed represented by $V$.

    Thus we can transfer properties back and forth between $G^\pm$ and $\vec{H}$.
    \begin{enumerate}[nosep, label=(\arabic*)]
        \item The number $\abs{G^\pm}$ of vertices of $G^\pm$ equals the number of edges, denoted $m$, of $\vec{H}$.
        \item The signed graph $G^\pm$ is connected if and only if $\vec{H}$ is connected except that $\vec{H}$ consists of a single pair of parallel edges.
        \item The signed graph $-G^\pm$ has a valid $p$-coloring if and only if the bidirected multigraph $\vec{H}$ has a proper $p$-edge-coloring.
        \item The multiplicity $\operatorname{mult}(-2, G^\pm)$ equals $\abs{G^\pm} - \rank V$, which in turn equals
        \[
            m - \rank V, \quad\text{where } V = \dset{\vec{v}(e)}{e \in E(\vec{H})}.
        \]
    \end{enumerate}

    For example, $\vec{H}$ could be the complete bipartite graph $K_{p,p}$ with all-positive incidences. Because $K_{p,p}$ has a proper $p$-edge-coloring, so does $\vec{H}$. Moreover one can compute $\rank V = 2p - 1$. This bidirected multigraph $\vec{H}$ and the corresponding signed graph $G^\pm$ establish \eqref{eqn:kpl-1} as follows:
    \[
        \frac{\abs{G^\pm}}{\operatorname{mult}(-2, G^\pm)} = \frac{m}{m - \rank V} = \frac{p^2}{p^2 - (2p-1)} = \left(\frac{p}{p-1}\right)^2.
    \]

    It suffices to show \eqref{eqn:kpl-2}, that is, for every connected bidirected multigraph $\vec{H}$ with $n$ vertices and $m$ edges, if $\vec{H}$ has a proper $p$-edge-coloring and $m \ge p^2$, then
    \[
        \frac{m}{m - \rank V} \ge \left(\frac{p}{p-1}\right)^2, \quad\text{where } V = \dset{\vec{v}(e)}{e \in E(\vec{H})}.
    \]

    Suppose that $c\colon E(\vec{H}) \to \sset{c_1, \dots, c_p}$ is a proper edge coloring of $\vec{H}$. One can deduce from \cref{def:proper-edge-coloring} the following properties.
    \begin{enumerate}[nosep, label=(\alph*)]
        \item For two edges $e_1$ and $e_2$ that intersect at a vertex $v$, if $\sigma(v, e_1) = \sigma(v, e_2)$, then $c(e_1) \neq c(e_2)$.\label{item:p-a}
        \item For two edges $e_1$ and $e_2$ that intersect at a vertex $v$, if $\sigma(v, e_1) \neq \sigma(v, e_2)$, then $c(e_1) = c(e_2)$, and any other edge incident to $v$ has to be parallel to either $e_1$ or $e_2$.\label{item:p-b}
        \item For a pair of parallel edges $e_1$ and $e_2$ that are incident to a vertex $v$, if $\sigma(v, e_1) \neq \sigma(v, e_2)$ and $c(e_1) = c(e_2)$, then $e_1$ and $e_2$ are the only two edges incident to $v$.\label{item:p-c}
    \end{enumerate}
    We break the rest of the proof into two cases.

    \paragraph{Case 1:} $\vec{H}$ has no parallel edges. Since $\vec{H}$ is connected, one can check that the set of vectors $\dset{\vec{v}(e)}{e \in E(T)}$, where $T$ is a spanning tree of $\vec{H}$, is linearly independent. Therefore
    \begin{equation} \label{eqn:kpl-3}
        \rank V \ge n - 1.
    \end{equation}

    We claim that the maximum degree of $\vec{H}$ is at most $p$. Suppose on the contrary that $k$ edges $e_1, \dots, e_k$ pairwise intersect at a vertex $v$, where $k > p$. Because $k > p \ge 3$, we derive from the contrapositive of \ref{item:p-b} that $\sigma(v, e_i)$ is the same for $i \in \sset{1, \dots, k}$, which contradicts with \ref{item:p-a}.

    The claim implies that $2m \le pn$. Together with $m \ge p^2$, we obtain
    \[
        \frac{m}{m - \rank V} \stackrel{\eqref{eqn:kpl-3}}{\ge} \frac{m}{m - (n - 1)} \ge \frac{m}{m - 2m/p + 1} \ge \left(\frac{p}{p-1}\right)^2.
    \]

    \paragraph{Case 2:} $\vec{H}$ has parallel edges. Let $e_1$ and $e_2$ be a pair of parallel edges. Since $\vec{H}$ is connected, there is a spanning tree $T$ that contains $e_1$. One can check that the set of vectors $\dset{\vec{v}(e)}{e \in E(T) \cup \sset{e_2}}$ is linearly independent. Therefore
    \begin{equation} \label{eqn:kpl-4}
        \rank V = n.
    \end{equation}

    We next use the discharging method to bound the average degree $2m / n$ of $\vec{H}$. Each vertex $v$ starts with the charge $d(v)$, that is, the degree of $v$. We claim that if $d(v) > p$ then exactly one of the following two situations happens.
    \begin{enumerate}[nosep, label=(\Roman*)]
        \item The sign $\sigma(v, e)$ is the same for every edge $e$ incident to $v$. \label{item:p-1}
        \item The edges incident to $v$ are precisely two pairs $(e_1, e_1')$ and $(e_2, e_2')$ of parallel edges satisfying $\sigma(v, e_1) \neq \sigma(v, e_2)$, and in particular $d(v) = 4$ and $p = 3$. \label{item:p-2}
    \end{enumerate}
    Indeed, suppose that $d(v) > p$, and suppose that situation \ref{item:p-1} does not happen, that is, there exist two edges $e_1$ and $e_2$ incident to $v$ such that $\sigma(v, e_1) \neq \sigma(v, e_2)$. Because $d(v) > p \ge 3$, we may further assume that $e_1$ and $e_2$ intersect at $v$ by replacing $e_1$ or $e_2$ with another edge incident to $v$ in case $e_1$ and $e_2$ are parallel. Because $d(v) \ge 4$, according to \ref{item:p-b}, situation \ref{item:p-2} must happen.

    To describe the discharging rules, we say that a pair of parallel edges $e_1$ and $e_2$ is \emph{monochromatic} if $c(e_1) = c(e_2)$, and \emph{colorful} otherwise, and for every vertex $v$ with $d(v) > p$, we call $v$ a \emph{type I} vertex when \ref{item:p-1} happens, and a \emph{type II} vertex when \ref{item:p-2} happens. The first discharging rule sends $1$ from every type I vertex $v$ to each of its neighbors $v'$ that are connected to $v$ through a monochromatic pair of parallel edges. The second discharging rule sends $1/2$ from every type II vertex $v$ to each of its two neighbors.

    We first consider the charge of each vertex after the first discharging rule is applied. Suppose that $v$ is a type I vertex, and $v'$ is an arbitrary vertex that is connected to $v$ through a pair of parallel edges $e_1$ and $e_2$. Since $\sigma(v, e_1) = \sigma(v, e_2)$, it must be the case that $\sigma(v', e_1) \neq \sigma(v', e_2)$, and so $v'$ is not a type I vertex. Thus $v$ does not receive any charge from its neighbors. According to \ref{item:p-a}, among all the edges incident to $v$, only the parallel ones can receive identical colors. Thus after $v$ sends out charges to its neighbors, its charge becomes exactly the number of distinct colors assigned to the edges incident to $v$, and so the charge of $v$ is at most $p$ now. Suppose in addition that $e_1$ and $e_2$ form a monochromatic pair, or in other words, $v'$ receives $1$ from $v$. According to \ref{item:p-c}, the only edges incident to $v'$ are $e_1$ and $e_2$, and so the charge of $v'$ is at most $3$ now.

    We then consider the charge of each vertex after the second discharging rule is applied. Notice that the second discharging rule only kicks in when $p = 3$. Suppose that $v$ is a type II vertex, and two pairs $(e_1, e_1')$ and $(e_2, e_2')$ of parallel edges connect $v$ respectively to $v_1$ and $v_2$ such that $\sigma(v, e_1) \neq \sigma(v, e_2)$. Without loss of generality, we may assume that $\sigma(v, e_1) = +1$ and $\sigma(v, e_2) = -1$. We have the four possibilities for $\sigma(v, e_1')$ and $\sigma(v, e_2')$. Once $\sigma(v, e_1')$ and $\sigma(v, e_2')$ are chosen, we can determine, according to \ref{item:p-a} and \ref{item:p-b}, which edges incident to $v$ receive identical or distinct colors. We summarize the four possibilities in \cref{fig:four-config}.
    \begin{figure}[t]
        \centering
        \begin{tikzpicture}[
            very thick, baseline=(v.base),
            neg/.pic = {\draw[fill=white] (0,0) circle (0.1); \draw (-0.1,0)--(0.1,0);},
            pos/.pic = {\draw[fill=white] (0,0) circle (0.1); \draw (-0.1,0)--(0.1,0); \draw (0,0.1)--(0,-0.1);},
        ]
            \node[vertex] (a) at (-2,0) {};
            \node[vertex] (b) at (0,0) {};
            \node[vertex] (c) at (2,0) {};
            \draw (a) node[left] {$v_1$};
            \draw (0,-.1) node[below] {$v$};
            \draw (c) node[right] {$v_2$};
            \path[color=blue] (a) edge[bend left=45] (b);
            \path[color=blue] (b) edge[bend left=45] (c);
            \path[color=blue] (a) edge[bend left=-45] (b);
            \path[color=red] (b) edge[bend left=-45] (c);
            \draw (-0.4,0.3) pic {pos};
            \draw (0.4,0.3) pic {neg};
            \draw (-0.4,-0.3) pic {pos};
            \draw (0.4,-0.3) pic {pos};
        \end{tikzpicture}\quad%
        \begin{tikzpicture}[
            very thick, baseline=(v.base),
            neg/.pic = {\draw[fill=white] (0,0) circle (0.1); \draw (-0.1,0)--(0.1,0);},
            pos/.pic = {\draw[fill=white] (0,0) circle (0.1); \draw (-0.1,0)--(0.1,0); \draw (0,0.1)--(0,-0.1);},
        ]
            \node[vertex] (a) at (-2,0) {};
            \node[vertex] (b) at (0,0) {};
            \node[vertex] (c) at (2,0) {};
            \draw (a) node[left] {$v_1$};
            \draw (0,-.1) node[below] {$v$};
            \draw (c) node[right] {$v_2$};
            \path[color=blue] (a) edge[bend left=45] (b);
            \path[color=blue] (b) edge[bend left=45] (c);
            \path[color=red] (a) edge[bend left=-45] (b);
            \path[color=blue] (b) edge[bend left=-45] (c);
            \draw (-0.4,0.3) pic {pos};
            \draw (0.4,0.3) pic {neg};
            \draw (-0.4,-0.3) pic {neg};
            \draw (0.4,-0.3) pic {neg};
        \end{tikzpicture}\quad%
        \begin{tikzpicture}[
            very thick, baseline=(v.base),
            neg/.pic = {\draw[fill=white] (0,0) circle (0.1); \draw (-0.1,0)--(0.1,0);},
            pos/.pic = {\draw[fill=white] (0,0) circle (0.1); \draw (-0.1,0)--(0.1,0); \draw (0,0.1)--(0,-0.1);},
        ]
            \node[vertex] (a) at (-2,0) {};
            \node[vertex] (b) at (0,0) {};
            \node[vertex] (c) at (2,0) {};
            \draw (a) node[left] {$v_1$};
            \draw (0,-.1) node[below] {$v$};
            \draw (c) node[right] {$v_2$};
            \path[color=blue] (a) edge[bend left=45] (b);
            \path[color=blue] (b) edge[bend left=45] (c);
            \path[color=blue] (a) edge[bend left=-45] (b);
            \path[color=blue] (b) edge[bend left=-45] (c);
            \draw (-0.4,0.3) pic {pos};
            \draw (0.4,0.3) pic {neg};
            \draw (-0.4,-0.3) pic {pos};
            \draw (0.4,-0.3) pic {neg};
        \end{tikzpicture}\quad%
        \begin{tikzpicture}[
            very thick, baseline=(v.base),
            neg/.pic = {\draw[fill=white] (0,0) circle (0.1); \draw (-0.1,0)--(0.1,0);},
            pos/.pic = {\draw[fill=white] (0,0) circle (0.1); \draw (-0.1,0)--(0.1,0); \draw (0,0.1)--(0,-0.1);},
        ]
            \node[vertex] (a) at (-2,0) {};
            \node[vertex] (b) at (0,0) {};
            \node[vertex] (c) at (2,0) {};
            \draw (a) node[left] {$v_1$};
            \draw (0,-.1) node[below] {$v$};
            \draw (c) node[right] {$v_2$};
            \path[color=blue] (a) edge[bend left=45] (b);
            \path[color=blue] (b) edge[bend left=45] (c);
            \path[color=red] (a) edge[bend left=-45] (b);
            \path[color=red] (b) edge[bend left=-45] (c);
            \draw (-0.4,0.3) pic {pos};
            \draw (0.4,0.3) pic {neg};
            \draw (-0.4,-0.3) pic {neg};
            \draw (0.4,-0.3) pic {pos};
        \end{tikzpicture}
        \caption{Four possible ways to color the edges incident to $v$ and to assign signs to the incidences of $v$.} \label{fig:four-config}
    \end{figure}
    For each possibility, it is easy to check that $v$ did not receive any charge from $v_1$ or $v_2$ when the first discharging rule was applied. Therefore the final charge of $v$ is $3$.

    We are left to decide the final charge of $v_1$ and $v_2$. Observe that for $i \in \sset{1,2}$ there are, au fond, only three possible ways, as shown in \cref{fig:three-config}, to color $e_i$ and $e_i'$ and to assign $\sigma(v_i, e_i')$ and $\sigma(v_i, e_i')$. For the first possibility, according to \ref{item:p-c}, the only edges incident to $v_i$ are $e_i$ and $e_i'$, and so the final charge of $v_i$ is $5/2$. For the other two possibilities, one can easily deduce that any other edge, say $e$, incident to $v_i$ satisfies $\sigma(v_i, e) = \sigma(v_i, e_i) = \sigma(v_i, e_i')$. If $d(v_i) \le 3$, then the final charge of $v_i$ is clearly $d(v_i) + 1/2$, which is at most $7/2$. If $d(v_i) \ge 4$, then $v_i$ is a type I vertex whose charge is at most $p = 3$ before the second discharging rule is applied, and so the final charge of $v_i$ is $7/2$.

    \begin{figure}[t]
        \centering
        \begin{tikzpicture}[
            very thick, baseline=(v.base),
            neg/.pic = {\draw[fill=white] (0,0) circle (0.1); \draw (-0.1,0)--(0.1,0);},
            pos/.pic = {\draw[fill=white] (0,0) circle (0.1); \draw (-0.1,0)--(0.1,0); \draw (0,0.1)--(0,-0.1);},
        ]
            \node[vertex] (a) at (-2,0) {};
            \node[vertex] (b) at (0,0) {};
            \draw (a) node[left] {$v$};
            \draw (b) node[right] {$v_i$};
            \path[color=blue] (a) edge[bend left=45] (b);
            \path[color=blue] (a) edge[bend left=-45] (b);
            \draw (-0.4,0.3) pic {pos};
            \draw (-0.4,-0.3) pic {neg};
        \end{tikzpicture}\quad%
        \begin{tikzpicture}[
            very thick, baseline=(v.base),
            neg/.pic = {\draw[fill=white] (0,0) circle (0.1); \draw (-0.1,0)--(0.1,0);},
            pos/.pic = {\draw[fill=white] (0,0) circle (0.1); \draw (-0.1,0)--(0.1,0); \draw (0,0.1)--(0,-0.1);},
        ]
            \node[vertex] (a) at (-2,0) {};
            \node[vertex] (b) at (0,0) {};
            \draw (a) node[left] {$v$};
            \draw (b) node[right] {$v_i$};
            \path[color=blue] (a) edge[bend left=45] (b);
            \path[color=red] (a) edge[bend left=-45] (b);
            \draw (-0.4,0.3) pic {pos};
            \draw (-0.4,-0.3) pic {pos};
        \end{tikzpicture}\quad%
        \begin{tikzpicture}[
            very thick, baseline=(v.base),
            neg/.pic = {\draw[fill=white] (0,0) circle (0.1); \draw (-0.1,0)--(0.1,0);},
            pos/.pic = {\draw[fill=white] (0,0) circle (0.1); \draw (-0.1,0)--(0.1,0); \draw (0,0.1)--(0,-0.1);},
        ]
            \node[vertex] (a) at (-2,0) {};
            \node[vertex] (b) at (0,0) {};
            \draw (a) node[left] {$v$};
            \draw (b) node[right] {$v_i$};
            \path[color=blue] (a) edge[bend left=45] (b);
            \path[color=red] (a) edge[bend left=-45] (b);
            \draw (-0.4,0.3) pic {neg};
            \draw (-0.4,-0.3) pic {neg};
        \end{tikzpicture}
        \caption{Three possible ways to color $e_i$ and $e_i'$ and to assign $\sigma(v_i, e_i)$ and $\sigma(v_i, e_i')$.} \label{fig:three-config}
    \end{figure}

    Since the discharging rules preserve the total charge, the average degree $2m/n$ is at most the maximum final charge, which is at most $p + 1/2$, which is less than $2p^2/(2p-1)$. Finally we obtain
    \begin{equation*}
        \frac{m}{m-\rank V} \stackrel{\eqref{eqn:kpl-4}}{=} \frac{m}{m-n} > \frac{1}{1-(2p-1)/p^2} = \left(\frac{p}{p-1}\right)^2. \qedhere
    \end{equation*}
\end{proof}

\section{Concluding remarks} \label{sec:discussion}

Extending Smith's classification \cite{S70} of connected graphs with spectral radius at most $2$, Cvetkovi\'c, Doob and Gutman~\cite[Theroem~3.8]{CDG82} characterized the connected graphs with spectral radius in $(2, \lambda')$, where $\lambda' = \sqrt{2+\sqrt5} \approx 2.05817$, which were later explicitly classified by Brouwer and Neumaier~\cite{BN89}. We naturally would like to extend \cref{thm:cgss} of Cameron et al.~\cite{CGSS76} beyond $\mathcal{G}(2)$.

\begin{problem}
    Classify all the connected graphs with smallest eigenvalues in $(-\lambda^*, -2)$. In particular, classify these graphs that have sufficiently many vertices.
\end{problem}

It is worth mentioning that Bussemaker and Neumaier~\cite[Theorem~2.5]{BN92} showed that $E_{2,6}$ (see \cref{fig:e2n}) is the only connected graph with smallest eigenvalue in $[-\lambda^1(E_{2,6}), -2)$, where $\lambda^1(E_{2,6}) \approx 2.00659$.

We can ask the same question for signed graphs, extending \cref{thm:cgss-signed} beyond $\mathcal{G}^\pm(2)$.

\begin{problem}
    Classify all the connected signed graphs with smallest eigenvalues in $(-\lambda^*, -2)$. In particular, classify these signed graphs that have sufficiently many vertices.
\end{problem}

Turning to spherical two-distance sets with two fixed angles, it is plausible to establish more instances of \cref{conj:main} by taking advantage of $\chi(G^\pm)$ in \cref{def:mph}. Denote $\mathcal{G}^\mp_p(\lambda)$ the family of signed graphs $G^\pm$ with $\chi(G^\pm) \le p$ and $\lambda^1(G^\pm) \le \lambda$. Observe that $\mathcal{G}^\mp_p(\lambda)$, just like $\G^\mp(\la)$, is still closed under taking subgraphs. We raise the following question, whose solution could possibly establish more instances of \cref{conj:main}.

\begin{problem} \label{prob:d}
    For every $p \in \mathbb{N}^+$, determine the set of $\lambda \in \mathbb{R}$ for which $\mathcal{G}^\mp_p(\lambda)$ has a finite forbidden subgraph characterization.
\end{problem}

Notice that $\mathcal{G}^\mp_1(\lambda)$ consists of (unsigned) graphs only, and $\mathcal{G}^\mp_2(\lambda)$ consists of signed graphs that are switching equivalent to (unsigned) graphs. Thus \cref{thm:forb1} essentially answers \cref{prob:d} when $p \in \sset{1,2}$.

\section*{Acknowledgements}

We thank Sebastian Cioaba for sending us a copy of \cite{H77a}, and Jing Zhang for pointing us to \cite{D13}.

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redpajama_set_name: RedPajamaArXiv