text:
\section{Introduction}
In this manuscript we study  the $b$--vector
$(b_1,b_2\ldots,b_d)$ of a chordal graph $G$ where $d$ is the largest cardinality of a clique in $G$. We consider non-complete graphs since in this case we conclude that $b_i=1$ for every $i=1,\ldots,d$.

If $c_i$ denotes the number of cliques of $G$ with $i$ vertices, then
the clique vector is given by  $c(G)= (c_1,c_2\ldots,c_d)$.
We note that the $c$--vector is the $f$--vector of the clique complex of $G$ shifted by one.
This vector is a classical invariant  of a graph and has  been intensively studied \cite{Frohmader,FrohmaderKKThm,H08,Zykov}.

The $b$--vector of $G$ given by $\mathbf{b}(G)=(b_1,b_2\ldots,b_d)$,  is a more recent numerical invariant \cite{H08} defined by the equation
$$
\sum^d_{i=1}b_i(x+1)^{i-1}=\sum^d_{i=1} c_i x^{i-1.}
$$

Goodarzi \cite{Goodarzi} showed that the vertex connectivity of $G,$ denoted by $\kappa$, is encoded in the $b$--vector as $b_1=1$ for $i\leq \kappa$ and $b_{\kappa+1}\neq 1.$
We extend this theorem by studying the remaining $b_i$ and relate them to the number of connected components  of $G$, $W(G\setminus Y),$ after the deletion of a set $Y$, with $i-1$ vertices.

We also relate $b_i$ to clique  domination (see Definition \ref{DefDominance}). Using this notion, we define the number $d_i(G)$, which  measures how many $i$--cliques are necessary to dominate every maximal clique of order at least $i$ in $G$.
In order to establish the relation between the $b_i$ and $d_i(G)$,
we need to introduce  $\tilde{\kappa}(G)$, which  is defined as the maximum  cardinality of the intersection of any pair of maximal cliques in $G$.

 Using these combinatorial invariants of $G$, we can state the main result regarding the structure of a chordal graph encoded by its $b$-vector.

\begin{theorem}[{See Corollary \ref{b_i} and Theorem \ref{k+1}}]\label{MainThm}
Let $G$ be a a chordal graph with vertex connectivity $\kappa.$
Then,
\begin{itemize}
\item[(a)]  $b_{i}=\sum_{|Y|=i-1} (W(G-Y)-1)+1$ for $1\leq i\leq \kappa +1$, where $Y\subseteq V(G)$;
\item[(b)]  $b_{i}<\sum_{|Y|=i-1} (W(G-Y)-1)+1$ for $\kappa+2\leq i\leq d$, where $Y\subseteq V(G)$;
\item [(c)] $b_i\le d_i(G)$ for every $i=1,\ldots,d$;
\item [(d)] $b_i=d_i(G)$ for every $i>\widetilde{\kappa}(G)$;
\item [(e)] $b_i\le b_j$  for every $\widetilde{\kappa}(G)<j\leq i$.
\end{itemize}
\end{theorem}

A key component of parts of the previous theorem is the use of the Betti numbers of the Stanley-Reisner ring associated to $\Delta(G),$ the simplicial complex of cliques of $G$. This  differs from Goodarzi's approach, since he only looked at the projective dimension of this ring.
In particular, we  reinterpret parts of  Theorem \ref{MainThm} in terms of these Betti numbers using Proposition \ref{beta} and the fact that $I_{\Delta(G)}$ has a $2$-linear resolution when $G$ is chordal.
This is given by  a formula for  $\beta_{i}(R/I_{\Delta(G)})$  in terms of  the $b$--vector (see Proposition \ref{beta y b}).

\begin{theorem}
Let $G$ be a a chordal graph with $n$ vertices and vertex connectivity $\kappa$. Let $\beta_{i}(R/I_{\Delta(G)})$ be the $i$-th  Betti number of the Stanley-Reisner ring $R/I_{\Delta(G)}$ of $\Delta(G)$.
Then,
\begin{itemize}
\item[(a)] $b_{i}=\beta_{n-i}(R/I_{\Delta(G)})+1$ for every $i=1,\ldots,\kappa+1$;
\item [(b)]  $ b_i<\beta_{n-i}(R/I_{\Delta(G)})$ for every $j\ge \kappa+2$.
\end{itemize}
\end{theorem}

Another key component of the proof of Theorem \ref{MainThm} is given by algebraic shifting. In particular, we compare the exterior, $\Delta(G)^e$, and symmetric, $\Delta(G)^s$, shiftings of $\Delta(G).$
As a consequence, we prove that $I_{\Delta^e}$ and $I_{\Delta^s}$ have the same graded Betti numbers for every simplicial complex $\Delta$ such that $I_\Delta$ has a linear resolution.
This gives a partial case of a conjecture of Aramova, Herzog and Hibi \cite[Conjecture 2.3]{AHH}. Furtheremore, we also prove this conjecture  for every $t$-skeleton of $\Delta(G)$, which recover a result by Murai \cite{Murai}.

The final key component for the main result is given by an explicit description   a combinatorial shifting for chordal graphs (see Definition \ref{combishiftingraph}).

\section{Background}

\subsection{Graph terminology}

 In this manuscript we consider a simple graph $G=(V(G),E(G))$ with set of vertices $V(G)$ and  edges $E(G)$. We also assume that $G$ is not the complete graph.
We now recall some concepts from graph terminology.

The \emph{order}  of $G$   is $n=|V(G)|$.
The \emph{degree} of a vertex $v\in V(G)$, denoted as  $\deg(v)$,  is the number of vertices in $G$ adjacent to $v$ and $N(v)$ denotes the set of neighbors of $v$ in $G$. We say that $G$ is  \emph{connected} if there is a path between any two vertices of $G$.
A  \emph{vertex-cut} of  $G$ is a set of vertices whose removal disconnects $G$. Every graph that is not complete has a vertex-cut.
The \emph{vertex connectivity} $\kappa=\kappa(G)$ of a graph $G$ is the minimum  cardinality of a
vertex-cut  and a graph is $k$\emph{--connected} if $k\le \kappa(G)$.
Given a vertex-cut $Y$ of $G$, we denote as $W(G-Y)$ the number of connected components in the graph $G-Y$.

Since the $b$--vector is defined in terms of its clique vector, we recall all the concepts necessary for our study.
A \emph{clique} is a subset of vertices of a graph such that its induced subgraph is complete and a $i$--clique is a clique of order $i$.
The \emph{clique vector} $\mathbf{c}(G)$ of a graph $G$ is a vector $(c_1,c_2,\ldots,c_d)$ in $\mathbb{N}^d$, where $c_i$ is the number of cliques in $G$ with $i$ vertices and $d$ is the largest cardinality of a clique in $G$.
A \emph{maximum} clique of a graph $G$ is a clique such that there is no clique with more vertices and the \emph{clique number} is the number of vertices in a maximum clique of $G$.
A \emph{maximal clique}  in $G$ is a clique which is not contained in any other clique of $G$ and we denote the set of maximal cliques of size $i$ by  $\mathcal{C}_i(G)$  for every $1\le i\le d$.

\begin{definition}\label{DefDominance}
We say that a clique $C$ of $G$ \emph{dominates} a clique $C'$ if $C\subset C'$. A \emph{dominating $i$--clique} of $G$ is a set of cliques of order $i$ in $G$ that dominates all maximal cliques of order at least $i$ in $G$.
For every $1\le i\le d$, we take
$$
d_i(G)=\min\{ |\mathcal{D}| \;|\: \mathcal{D} \hbox{ is a dominating }i\hbox{-clique of } G    \}.
$$
We say that a dominating $i$--clique $\mathcal{D}$
is   minimum  if $|\mathcal{D}|=d_i(G)$.
\end{definition}

For a simplicial complex $\Delta$ on a set of vertices $V$ and $Y\subseteq V$, we denote as $\Delta_{|_Y}$ the simplicial subcomplex of $\Delta$ restricted in $Y$ and as $W(\Delta-Y)$ the number of connected components in $\Delta_{|_{V\setminus Y}}$.  The $t$--skeleton of $\Delta$ is given by $\Delta^{(t)}=\{\tau\in\Delta\;|\; \dim(\tau)\leq t\}$, in particular $\Delta(G)^{(1)}=G.$ The connectivity of a simplicial complex $\Delta$ is defined as the connectivity of its $1$--skeleton. The set of cliques in $G$ forms simplicial complex  $\Delta(G)$, known as the \emph{clique complex} of $G$. Then the well-known $f$\emph{-vector} of $\Delta(G)$ is exactly the clique vector of $G$.

A graph is \emph{chordal} if every cycle of length at least $4$ has a chord i.e., an edge that is not part of the cycle but connects two vertices of the cycle.
Given the clique vector $\mathbf{c}(G)$ of a chordal graph $G$, we define the vector $b$--vector of $G$, $\mathbf{b}(G)=(b_1,b_2,\ldots,b_d)$, defined as
\begin{equation}\label{eq1}
\sum\limits_{i=1}^{d}b_i(x+1)^{i-1}=\sum\limits_{i=1}^{d}c_ix^{i-1}.
\end{equation}

We seek to show that as $\mathbf{c}(G)$ give us the number the $i$--cliques of $G$,   $\mathbf{b}(G)$  also give us  structural information if $G$ is chordal.

\subsection{Free resolutions and Stanley-Reisner rings}
In this subsection we consider a polynomial ring $R=K[x_1,\ldots,x_n]$ as a $\mathbb{N}^n$--graded ring with $\deg(x_i)=e_i$
where $e_i$ is the vector with one in the  $i$--th entry and zeros elsewhere.
We take $\mathfrak{m}=(x_1,\ldots,x_n).$
Let $M$ be a graded finitely generated $R$--module.
Then, there exists a resolution by graded free modules
$$
0\to  F_p \FDer{\varphi_{i-1}} F_{i-1}\to \ldots \FDer{\varphi_0} F_o\to M\to 0,
$$
where $\varphi_i$ is represented by a matriz with homogebeous entries in $\mathfrak{m}.$
We can  write $F_i = \bigoplus_j R(-j)^{\beta_{i,\alpha} (M)}$, where $R(-\alpha)$ denotes a rank one free module with an  homogeneous generator in degree $-\alpha$.
By The Hilbert Syzygy Theorem, $p\leq n$.
The projective dimension of $M$ is defined by $\operatorname{pd}(M)=p.$ By the Auslander-Buchsbaum formula, we have that
$\Depth(M)=n-\operatorname{pd}(M)$.
The numbers $\beta_{i,j}(M)$ are important invariants for $M$, with a vast number of applications in algebra, geometry, and topology.
The Castelnuovo-Mumford regularity is defined by $\operatorname{reg}(M)=\max\{j\;|\; \beta_{i,\alpha}(M) \neq 0\hbox{ with }|\alpha|= i+j\}$.

We say that $M$ has \emph{$t$--linear resolution} if every homogeneous minimal generator of $M$ has degree $t$ and $\beta_{i.i+j}(M)=0$ for $j\neq t$.

We refer Peeva's book on free resolutions \cite{PeevaGrSyz} and Eisenbud's book on syzygies \cite{EisenbudSyzygy}
for more details about free resolutions and their geometry.

An ideal generated by monomials is square free, if it is radical.
There is a well-known  bijection between the squarefree monomial ideals in $R$ and  simplicial complexes in $n$.

Given a simplicial complex $\Delta$, the square free monomial associated to $\Delta$ is defined by
$$I_{\Delta} = \left( x^\sigma\; |\;\sigma  \notin \Delta \right).$$
The quotient $R / I_{\Delta}$ is called the Stanley-Reisner ring of $\Delta$.

Given a square-free monomial ideal $I$,  the simplicial complex associated to $I$ is defined by
$$\Delta = \left\lbrace \sigma \subseteq [n] \; | \; x^\sigma \notin I \right\rbrace.$$

We refer to the book by Miller and Sturmfeels  on combinaturial commutative algebra \cite{MillerSturmfels} and the survey by Francisco, Mermin and Schweig on Stanley-Reisner theory \cite{SurveySR}  for more details about these rings.

Hochster's Formula \cite[Theorem 5.1]{HochsterFormula} is an important result that connects the topology of simplicial comlexes to free resolutions, which plays a key role in this manuscript.
Let $W=\mbox{\rm{supp}} (\alpha).$
Then,
$$\beta_{i+1,\alpha}(R/I_\Delta)=\beta_{i,\alpha}(I_\Delta)= \tilde{H}^{|W|-i-2}(\Delta_{|_W }; K)$$
for every $i\in\mathbb{N}$ and $\alpha\in\mathbb{N}^n.$

\section{Threshold Graphs}
In this section, we focus on the $b$--vector of threshold graphs. This is a key part of our strategy since the $b$--vector determines a unique threshold graph, and therefore, its graph properties. To expand our result to chordal graphs we use results from shifting theory in Subsection \ref{SubsectionShift}.

\subsection{Definition and basic properties}
A threshold graph is a graph that can be constructed from a one-vertex graph by the following two operations: addition of one isolated vertex, denoted by $D$,  and addition of a vertex connected to all other vertices, denoted by $S$. We set the convention of always start with an $S$--operator. Thus, there is a bijection between threshold graphs and words on the alphabet $\{S,D\}$ beginning with an $S$ reading from left to right.

\begin{figure}[htb]\label{FigExThreshold}
\begin{center}
 \includegraphics[width=.2\textwidth]{threshold.pdf}
\caption{Graph corresponding to the word SDSDDS }\label{threshold}
\end{center}
\end{figure}

We consider a vertical line before every $S$ in the word corresponding to the threshold graph $T$, thus breaking the word into subwords.
Goodarzi \cite{Goodarzi} showed that the entry
 $b_i$ of the $b$--vector  $\mathbf{b}(G)$ is the length of the $(d-i+1)$--th subword for every $i=1,\ldots,d$. As a consequence, there is a bijection between $b$--vectors and threshold graphs.
 For instance, the graph in Figure \ref{FigExThreshold}, which has the word $SDSDDS$,
 breaks to $|SD|SDD|S$ and $(b_1,b_2,b_3)=(1,3,2)$.

Given a threshold graph $T$, we recall that $\mathcal{C}_i(T)$ is the set of maximal cliques of size $i$ in $G$ and we denote as  $C_i^s$ the $i$--clique of $T$  conformed by all vertices corresponding to all the letters $S$ in the  $(d-j+1)$--th subwords, $1\le j\le i$.
Many properties of a threshold graph can be read off from its word. We collect some simple observations.

\begin{observation}\label{obs}
Let $T$ be a threshold graph with clique number $d$, then the following hold:
\begin{itemize}
  \item [(a)] The number of times that $S$ appears in $T$ is the clique number of $T$.
  \item [(b)] The connectivity of $T$ is the number of consecutive $S$ appearing at the end of the word $T$.
  \item [(c)]   There is only one minimum vertex-cut in $T$.
  \item [(d)] For any vertex-cut $Y$ of $T$, the graph $T-Y$ has at most one component with at least $2$ vertices.
  \item [(e)] $\mathcal{C}_i(T)\cup C_i^s$ is the unique  minimum  dominating $i$--clique of $T$ for every $1\le i\le d$.
\end{itemize}
\end{observation}

We point out that there are others characterization of threshold graphs. We recall one that is useful in our study.
\begin{theorem}[\cite{MahadevPeled}]\label{Mahadev}
A graph is threshold if and only if it does not contains the graphs $G_1$, $G_2$ or $G_3$ of Figure \ref{G_1G_2G_3}
as an induced subgraph.
\end{theorem}
\begin{figure}[htb]
\begin{center}
 \includegraphics[width=.4\textwidth]{G_1G_2G_3.pdf}
\caption{ }\label{G_1G_2G_3}
\end{center}
\end{figure}

\subsection{$b$--vectors of threshold graphs}

We start this section with a result that establishes a strong relation between the $b$--vector and the graphs structure.

\begin{proposition}\label{dominate}
Let $T$ be a threshold graph  with  vertex connectivity $\kappa$.
 Then,
\begin{itemize}
\item [(a)]  $b_1=b_2=\ldots=b_{\kappa}=1$;
 \item [(b)] $b_i-1=|\mathcal{C}_i(T)|$ for every $\kappa+1\le i\le d-1$ and $b_d=|\mathcal{C}_d(T)|$;
 \item [(c)] $b_i=d_i(T)$ for every $i=1,\ldots,d$;
 \item [(d)] $b_{\kappa+1}=W(T-Y)$ for a minimum vertex-cut $Y$.
 \end{itemize}
\end{proposition}
\begin{proof}$ $
\begin{itemize}
\item[(a)] It follows from Observation \ref{obs}(b).

\item[(b)]
Let $\kappa+1\le i\le d-1$.
Then, each maximal clique of size $i$ in $T$ is forming by a vertex in $T$ corresponding to a letter $D$ in the $(d-i+1)$--th subword together with the vertices in $T$  corresponding to all the letters $S$ in the  $(d-j+1)$--th subwords for every $1\le j\le i-1$.
Since the $(d-i+1)$--th subword has exactly one $S$ and $b_i$ corresponds to the cardinality of letters of its $(d-i+1)$--th subword, we obtain that $b_i-1=|\mathcal{C}_i(T)|$.
Finally, if we put the letter $D$ instead $S$ in the first position of the word corresponding to $T$, we  obtain the same graph $T$. Hence, the previous argument holds for $i=d$ obtaining that $b_d=|\mathcal{C}_d(T)|$.

\item[(c)] We note that $\mathcal{C}_i(T)\cup C_i^s$ is the unique  minimum  dominating $i$--clique of $T$ by Observation \ref{obs}(e). If $1\le i\le \kappa$, then the set $\mathcal{C}_i(T)=\emptyset$ and the clique $C_i^s$ is a minimum  dominating $i$--clique of $T$ proving that $d_i(T)=1$ for every $1\le i\le \kappa$. We conclude that $b_i=d_i(T)=1$ for every $1\le i\le \kappa$  by (a).
If $\kappa+1\le i\le d$, we have that $b_i-1=|\mathcal{C}_i(T)|$ for every $\kappa+1\le i\le d-1$ and $b_d=|\mathcal{C}_d(T)|$ by (b).
Since  $\mathcal{C}_i(T)\cup C_i^s$ is a minimum  dominating $i$--clique in $T$  by Observation \ref{obs}(e), it follows that $b_i=d_i(T)$ for every $\kappa+1\le i\le d$, because  $i=d$ implies $C_i^s\in \mathcal{C}_i(T)$.

\item[(d)] By Observation \ref{obs}(b)(c), there is only one minimum vertex-cut $Y$ in $T$, which is formed by the vertices of $T$ corresponding to all consecutive $S$ appearing at the end of the word corresponding to $T$. Therefore,  $b_{\kappa+1}$ is the number of components in $T-Y$.
\end{itemize}
\end{proof}

One of the main objectives of this paper is to extend the previous proposition to chordal graphs and  to compare the $b$--vector of a chordal graph $G$ with the Betti numbers of $\Delta(G)$.
We end this subsection with a result that further the relation of the  $b$--vector with connectivity.

\begin{proposition}\label{k+2}
Let $T$ be a threshold graph with vertex connectivity $\kappa$.
  Then,
$$
    b_{i+1}<\sum\limits_{\substack{|Y|=i}} (W(T-Y)-1)
  $$ for every $\kappa+1\le i\le d-1$ where $Y\subseteq V(T)$.
\end{proposition}
\begin{proof}
We observe that if $Y\subseteq V(T)$ is not a vertex-cut of $T$, then $W(T-Y)-1=0$. Hence the sum $\sum\limits_{\substack{|Y|=i}} (W(T-Y)-1)$ only consider vertex-cuts $Y$ of $T$.
 Let denote as $B_i$ the set of vertices in $T$ that correspond to the letters in the $(d-i+1)$--th subword  for every $i=1,\ldots,d$ (i.e., $|B_i|=b_i$). By Proposition \ref{dominate}(a), for every $i=1,\ldots,\kappa$ the set  $B_i$ has only one vertex, say $x_i$,  and by Observation \ref{obs}(c) the only  minimum vertex-cut in $T$  is the set $\{x_i\}_{i=1}^{\kappa}$.

   Let $V_{\kappa+\ell}$ be the number of all vertex-cuts of $T$ with cardinality $\kappa+\ell$, then clearly $V_{\kappa+\ell}\le \sum\limits_{\substack{|Y|=\kappa+\ell}} (W(T-Y)-1)$ since $W(T-Y)-1\ge 1$ for every vertex-cut $Y$ of $T$ with cardinality $\kappa+\ell$. Therefore, it is enough to show that $b_{\kappa+1+\ell}< V_{\kappa+\ell}$ for every $1\le\ell\le d-\kappa-1$.

   Suppose that $T$ has order $n$, let $a\in B_{\kappa+1}$ be such that its correspond letter is $D$ and consider the set $\bigcup\limits_{j=\kappa+1}^{d} B_j-\{a\}$ of cardinality $n-\kappa-1$. As $d\le n-1$ since $b_{\kappa+1}\ge 2$, we have that $\ell<n-\kappa-1$. Then, for each $L\subset\bigcup\limits_{j=\kappa+1}^{d} B_j-\{a\}$ with cardinality $|L|=\ell$ we have that $Y=\{x_j\}_{j=1}^{\kappa}\cup\{L\}$ is a vertex-cut of $T$ with cardinality $\kappa+\ell$ since vertex $a$ is isolate in $T-Y$. Therefore, there are at least ${n-\kappa-1\choose \ell}$ vertex-cuts of $T$ with cardinality $\kappa+\ell$ concluding that ${n-\kappa-1\choose \ell}\le V_{\kappa+\ell}$. On the other hand, we notice that $b_{\kappa+1+\ell}<n-\kappa-1$ for every   $1\le \ell\le d-\kappa-1$, otherwise we would have that $b_j=1$ for every $j=1,\ldots,\kappa+1$, a contradiction since $b_{\kappa+1}\ge2$. Thus, as $n-\kappa+1\le {n-\kappa-1\choose \ell}$ for every $1\le \ell< n-\kappa-1$,  we conclude that $b_{\kappa+1+\ell}<V_{\kappa+\ell}$.
\end{proof}

\section{Algebraic tools}

\subsection{Betti numbers and connectivity}
 The following result has already been pointed out by several authors \cite[Theorem 6]{Goodarzi}
 \cite[Corollary 1.2]{KatzmanBettiGraph}. We include a proof in the contexts we need for the sake of completeness.
\begin{proposition}\label{beta}
Let $G$ be a graph of order $n$ and let $\Delta=\Delta(G)$.
Then,
$$
 \beta_{i,i+1}(R/I_\Delta)=\sum_{|Y|=n-i-1} (W(G-Y)-1),
$$  for every $1\le i\le n$, where $Y\subseteq V(G)$.
  As a consequence,
$G$ is $k$--connected if and only if
$\beta_{i,i+1}(R/I_\Delta) =0$ for all $i\geq n-k.$ In particular,
$$
\kappa(G) =\max\{ k\; | \; \beta_{i,i+1}(R/I_\Delta) =0\hbox{ for all }i\geq n-k\}.
$$
\end{proposition}
\begin{proof}
From Hochster's Formula \cite[Theorem 5.1]{HochsterFormula}, we have that
\begin{align*}
\beta_{i,i+1}(R/I_\Delta)=\beta_{i-1,i+1}(I_\Delta)&=\sum_{|Y|=i+1}
\dim_K \widetilde{H}^0 (\Delta_{|_Y};K)\\
&=\sum_{|Y|=i+1}  (W(G_{|_Y})-1)\\
&=\sum_{|Y|=n-i-1} (W(G- Y)-1)\\
&=\sum_{|Y|=n-i-1}  (W(G- Y)-1).
\end{align*}
The rest follows from observing that $\beta_{i,i+1}(Y/I_\Delta)=0$ if and only if there is no vertex-cut $Y$ such that $|Y|=n-i-1.$
\end{proof}

The previous result implies the following result, which is already known  \cite{KatzmanBettiGraph,Goodarzi}.

\begin{corollary}[{\cite[Theorem 6]{Goodarzi}}]\label{CorDepth}
Let $G$ be a graph and set $\Delta=\Delta(G).$
Then, $$\Depth(R/I_\Delta)\leq \kappa(G)+1.$$
Furthermore, the equality is reached if $G$ is chordal.
\end{corollary}

\subsection{Shifting theory}\label{SubsectionShift}
In order to continue our study of $b$--vectors,  we need to recall a few terms and properties of shifting theory for simplicial complex.

\begin{definition}
A simplicial complex $\Delta$ on $[n]$ is shifted if, for $\sigma\in\Delta, i\in \sigma,$ and
$j\in [n]$ with
$j>i,$ one has $(\sigma\setminus \{ i\})\bigcup\{ j\}\in\Delta$.
\end{definition}

\begin{definition}
A shifting operation on $[n]$ is a map which associates each simplicial
complex $\Delta$ on $[n]$ with a simplicial complex $\operatorname{Shift}(\Delta)$ on $[n]$ and which satisfies the following conditions:
\begin{itemize}
\item[(a)]  $\operatorname{Shift}(\Delta)$ is shifted;
\item[(b)] $\operatorname{Shift}(\Delta)=\Delta$ if $\Delta$ is shifted;
\item[(c)] $f(\Delta) = f(\operatorname{Shift}(\Delta))$;
\item[(d)] $\operatorname{Shift}(\widetilde{\Delta})\subseteq\operatorname{Shift}(\Delta)$ if $\widetilde{\Delta}\subseteq\Delta$ and $\widetilde{\Delta}^{(0)}=\Delta^{(0)}$.
\end{itemize}
\end{definition}

\begin{definition}
We say that a shifting operation $\operatorname{Shift}(-)$ is compatible with Alexander Duality if
$\operatorname{Shift}(\Delta^\vee)=\operatorname{Shift}(\Delta)^\vee$
\end{definition}

There are several well-known shifting operations. Among these operations, the exterior shifting $\Delta^e$ and the symmetric shifting $\Delta^s$ stand up.
We point out that $\Delta^s$ is only defined in characteristic zero.
 We omit the definitions as its not needed for our purposes. However, we refer the interested reader in the book by Herzog and Hibi \cite[Chapter 11]{BookMonomial}.

 We recall a few properties for the exterior and symmetric shifting.

\begin{theorem}[{\cite[Theorem 11.4.1]{BookMonomial}}]\label{ThmShiftProp}
Let $\Delta$ be any simplicial complex.
The following statements holds:
\begin{enumerate}
\item $\Depth(R/I_\Delta)=\Depth(R/I_{\Delta^e})$.
\item $(\Delta^e)^\vee=(\Delta^\vee)^e$
\item $\operatorname{reg}(R/I_\Delta)=\operatorname{reg}(R/I_{\Delta^e})$.
\end{enumerate}
If the ground field has characteristic zero, then the same statements hold for the symmetric shifting.
\end{theorem}

We now mention a conjecture that relates the Betti numbers of algebraic and symmetric shifting.

\begin{conjecture}[{\cite[Conjecture 2.3]{AHH}}]\label{Conj}
Let $\Delta$ be a simplicial complex and suppose that $\Char(K)=0.$
Then,
$$\beta_{i,j}(I_{\Delta^s})\leq \beta_{i,j }(I_{\Delta^e}).$$
\end{conjecture}

In order to study the previous conjecture and the  $b$-vector of a chordal graph, we need to recall the definition and properties of the $h$-vector of a simplicial complex.

\begin{definition}
Let $\Delta$ be a simplicial complex of dimension $d-1$ and
let $f(\Delta)=(f_{-1},\ldots,f_{d-1})$ be the $f$-vector of $\Delta$.
The $h$-vector of $\Delta,$ $h(\Delta)=h_0,\ldots,h_d)$, is defined by
$$
\sum^d_{i=0}h_it^i=\sum^d_{i=0} f_{i-1} t^i (1-t)^{d-i}.
$$
\end{definition}

\begin{remark}\label{h-vector f-vector}
We have that
$$
h_j=\sum^j_{i=0}(-1)^{j-i}\binom{d-i}{j-i} f_{i-1} \quad \& \quad f_{j-1}=\sum^j_{i=0} \binom{d-i}{j-i} h_i.
$$
\end{remark}

We now recall a result  that would imply that the $f$-vector of certain simplicial complexes determine its Betti numbers.

\begin{theorem}[{\cite[Corollary 3.4]{H10}}]\label{ThmBettiH}
If $I_\Delta$ has a $t$-linear free resolution,
then
$$\beta_i(R/I_\Delta)=\sum^{t+i}_{\ell=0} (-1)^{\ell+i+1} h_{t+i-\ell} \binom{n-d}{\ell}$$
\end{theorem}

We  are able to answer Conjecture \ref{Conj} for ideals with a linear free resolution.

\begin{proposition}\label{PropPreservationBetta}
Let $\Delta$ be a simplical complex such that $I_\Delta$ has a $t$-linear free resolution. Then,
$\beta_{i,j}(R/I_\Delta)=\beta_{i,j}(R/I_{\Delta^e})$.
Furthermore, If $\Char(K)=0,$ then
$$
\beta_{i,j}(R/I_{\Delta^s})=\beta_{i,j}(R/I_{\Delta})=\beta_{i,j}(R/I_{\Delta^e}).
$$
\end{proposition}
\begin{proof}
Since $I_\Delta$ has a free $t$-linear reolution,
$I_{\Delta^s}$ and $I_{\Delta^e}$ also have a free resolution by Theorem \ref{ThmShiftProp}.
We conclude that
$$
\beta_{i,j}(R/I_{\Delta^s})=\beta_{i,j}(R/I_{\Delta})=\beta_{i,j}(R/I_{\Delta^e})=0.
$$
for $j\neq i+t-1$.
In addition,
$\beta_{i}(R/I_{\Delta^s})=\beta_{i,j}(R/I_{\Delta^s})$,
$\beta_{i,j}(R/I_{\Delta})=\beta_{i,j}(R/I_{\Delta})$, and
$\beta_{i,j}(R/I_{\Delta^e})=\beta_{i,j}(R/I_{\Delta^e}).$
Since shifting preserves $f$-vector, it also preserves $h$-vectors. Then, exterior and symmetric shifting preserve total Betti numbers in this case by Theorem \ref{ThmBettiH}.
Hence,
$$
\beta_{i,j}(R/I_{\Delta^s})=\beta_{i,j}(R/I_{\Delta})=\beta_{i,j}(R/I_{\Delta^e}).
$$
for $j=i+t-1.$
\end{proof}

As a consequence of Theorem \ref{ThmBettiH} and Proposition \ref{beta}, we prove that shifting preserves  numbers related to connectivity for chordal graphs.

\begin{corollary}\label{CorConnectivity}
 Let  $G$ be a chordal graph with vertex connectivity $\kappa$ and set $\Delta=\Delta(G).$
Then,
$$
\sum_{|Y|=\kappa} (W(\Delta- Y)-1)=
\sum_{|Y|=\kappa} (W(\Delta^e- Y)-1),
$$ where $Y\subseteq V(G)$.
Furthermore, if $\Char(K)=0$ ,
then
$$
\sum_{|Y|=\kappa} (W(\Delta- Y)-1)=
\sum_{|Y|=\kappa} (W(\Delta^s- Y)-1).
$$
\end{corollary}
\begin{proof}
This follows immediately from Propositions \ref{beta} and \ref{PropPreservationBetta}.
\end{proof}

We now recall an structural result that characterizes clique complexes of threshold graphs.

\begin{proposition}[{\cite[Theorem 2]{Carly}}]\label{TeoCarly}
The graph $G$ is threshold if and only if  $\Delta(G)$ is shifted.
\end{proposition}

Thanks to the previous result, we are able to show that the exterior and symmetric shifting yield the same complex for clique complexes of chordal graphs.

\begin{theorem}\label{TeoSameShift}
Let $G$ be a chordal graph. Suppose that $\Char(K)=0$.
Then $\Delta(G)^e=\Delta(G)^s.$
\end{theorem}
\begin{proof}
Since $G$ is a chordal graph, $I_{\Delta(G)}$ has a $2$--linear resolution.
Then,  $I_{(\Delta(G))^e}$ and $I_{(\Delta(G))^s}$ have a $2$--linear resolution.
Then,   $I_{(\Delta(G))^e}$ and $I_{(\Delta(G))^s}$ are the monomial edge ideals of certain graph.
As a consequence,
 $(\Delta(G))^e$ y $(\Delta(G))^s$ are clique complexes of certain graphs, say $G_1$ and $G_2$.
 By Theorem \ref{TeoCarly}, $G_1$ and $G_2$ are threshold graphs.
We obtain that
$$
f(\Delta(G_1))=f((\Delta(G))^e)=
f(\Delta)
=
f((\Delta(G))^e)=f(\Delta(G_2)).
$$
Then the clique vector of $G_1$ and $G_2$ are equal, concluding that $G_1$ and $G_2$ have the same $b$-vector, so $G_1=G_2$.
Hence
$\Delta(G_1)=\operatorname{Shift}_1(\Delta(G))=\operatorname{Shift}_2(\Delta(G))=\Delta(G_2).$
\end{proof}

As a consequence of Theorem \ref{TeoSameShift}, we show  Conjecture \ref{Conj} for clique complex of chordal graphs. We point out that the fact that $G^e=G^s$ was previously know \cite{Murai}.

\begin{proposition}\label{Prop e=s}
Suppose that $\Char(K)=0$. Then $\Delta(G)^e=\Delta(G)^s$ for every chordal graph.
As a consequence,
$(\Delta(G)^{(i)} )^e=(\Delta(G)^{(i)})^s$ and Conjecture \ref{Conj} holds for
$\Delta(G)^{(i)}$
for every $i\in\mathbb{N}$.
\end{proposition}
\begin{proof}
This is an immediate consequence from Theorem \ref{TeoSameShift}, and the fact that $\operatorname{Shift}(\Delta^{(i)})=(\operatorname{Shift}(\Delta))^{(i)}$ for every shifting operation $\operatorname{Shift}(-)$ and $i\in\mathbb{N}.$
\end{proof}

\section{Chordal graphs}

In this section we prove our main results regarding chordal graphs. We start by recalling the definition and properties of a perfect elimination order.

A vertex $v$ of a graph $G$ is called \emph{simplicial} if its adjacency set $N(v)$ induces a clique. An \emph{elimination ordering} $\sigma$ of a graph $G$ is a bijection $\sigma:\{1,\ldots,n\}\to V(G)$. Thus, $\sigma(i)$ is the $i$th vertex in the elimination ordering and for $v\in V(G)$, $\sigma^{-1}(v)$ gives the position of $v$ in $\sigma$. A \emph{perfect elimination ordering} (PEO) is an
elimination ordering  $\sigma=(v_1,v_2,\ldots,v_n)$ where $v_i$ is a simplicial vertex in the
subgraph induced by $\{v_i,v_{i+1},\ldots,v_n\}$, $1\le i\le n$. Given a PEO $\sigma$ of a graph $G$, the \emph{monotone adjacency set} of $v$, denoted $N_{\sigma}(v)$, is given by
$$N_{\sigma}(v):=\{u\in N(v) \mbox{ }\vert \mbox{ } \sigma^{-1}(v)<\sigma^{-1}(u)\}$$ and denote $n_{\sigma}(v):=|N_{\sigma}(v)|$. For a
 maximal clique $C$ of $G$, let define  $$s(C):=\{x\in C \mbox{ }\vert \mbox{ } N_{\sigma}(x)\not\subseteq C\}.$$ For any set of vertices $U=\{u_{1},u_{2},\ldots,u_r\}$ of $G$, we say that $U$ is \emph{well ordered} if $\sigma^{-1}(u_{i})<\sigma^{-1}(u_{j})$ for every  $i,j\in \{1,\ldots,r\}$ with $i<j$.

As shown below in Theorem \ref{Dirac}, every nontrivial chordal graph has at least two
simplicial vertices.

\begin{theorem}[{\cite{D61}}]\label{Dirac}
Every chordal graph has a simplicial vertex. If $G$ is not
the complete graph, then it has two nonadjacent simplicial vertices.
\end{theorem}

We know recall a characterization of chordal graphs related to PEO.

\begin{theorem}[{\cite{FulkersonGross}}]\label{Fulkerson}
A graph is chordal if and only if it has a PEO.
\end{theorem}

The next result shows the existence of a special PEO satisfying properties that are useful for the rest of the paper.

\begin{lemma}\label{K_d}
Let $G$ be a chordal graph of order $n$, let $K=\{x_1,x_2\ldots,x_k\}$  be a maximal clique of $G$ and let $C$ be any maximal clique of $G$. Then there exists a PEO  $\sigma$ of $G$ such that:
\begin{itemize}
  \item [(a)]  $\sigma^{-1}(u)<\sigma^{-1}(x)$ for every $x\in K$ and every $u\in V(G)-K$, moreover $\sigma^{-1}(x_i)=n-i+1$ for every $i=1,\ldots,k$;
  \item [(b)]  $\sigma^{-1}(u)<\sigma^{-1}(v)$ for every $u\in C-s(C)$ and every $v\in s(C)$;
  \item [(c)] if $|s(C)|<i\le |C|$, then there exists a unique vertex $u\in C-s(C)$ such that $n_{\sigma}(u)=i-1$;
   \item [(d)] if $C'$ is another maximal clique of $G$ such that $C'\cap C\neq\emptyset$, then $\sigma^{-1}(u)<\sigma^{-1}(v)$ for every $u\in C\setminus C'$ and every $v\in C\cap C'$ or   $\sigma^{-1}(u)<\sigma^{-1}(v)$ for every $u\in C'\setminus C$ and every $v\in C\cap C'$.
\end{itemize}

\end{lemma}
\begin{proof}

We construct a  PEO  $\sigma$ of $G$ as follows.  Let $\mathcal{C}$ be the set of maximal cliques of $G$ and suppose that $|\mathcal{C}|=r$.  There exists two nonadjacent simplicial vertices in $G$ by Theorem \ref{Dirac}. Then, one of them is a vertex $u_1\in V(G)-K$. Suppose that $u_1\in C_1$ for some $C_1\in \mathcal{C}\setminus\{K\}$. Set $P_1=\{u\in C_1 \mbox{ }\vert \mbox{ } u \mbox{ is simplicial in } G\}$.
We note that the graph $G_2=G-P_1$ is chordal and  for every $C\in \mathcal{C}-{C_1}$, $C$ is a maximal clique in $G_2$. By Theorem \ref{Dirac}, there exists a vertex $u_2\in V(G_2)-K$ simplicial in $G_2$. Suppose that $u_2\in C_2$ for some $C_2\in \mathcal{C}\setminus\{K,C_1\}$.  Let $P_2=\{u\in C_2 \mbox{ }\vert \mbox{ } u \mbox{ is simplicial in } G_2\}$. Proceeding in this way for every $i=3,4,\ldots,r-1$, we obtain that
the graph  $G_i=G_{i-1}-P_{i-1}$ is chordal. In addition,  $C$ is a maximal clique in $G_i$ for every $C\in \mathcal{C}\setminus\{C_1,C_2,\ldots,C_{i-1}\}$.
 Then, by Theorem \ref{Dirac}, there exists a vertex $u_i\in G_i-K$ simplicial in $G_i$. Suppose that $u_i\in C_i$ for some $C_i\in \mathcal{C}-\{K,C_1,C_2,\ldots,C_{i-1}\}$. Set $P_i=\{u\in C_i \mbox{ }\vert \mbox{ } u \mbox{ is simplicial in } G_i\}$. We define $\sigma$ as follows. For every $i,j=1,\ldots,r-1$ with $i<j$, every $p_i\in P_i$ and every $p_j\in P_j$, we take  $\sigma^{-1}(p_i)<\sigma^{-1}(p_j)$ and  $\sigma^{-1}(x_i)=n-i+1$ for every $i=1,\ldots,k$. Then, $\sigma$ is a PEO of $G$ by construction.
We now show that $\sigma$ satisfies the condition (a)-(d).
\begin{itemize}
  \item [(a)]  It follows by the construction of $\sigma$.

  \item [(b)] We notice that $\mathcal{C}=\{C_1,C_2\ldots,C_r\}$ where $C_r=K$. By construction of $\sigma$, the sets $S_i$ and $s(C_i)$ are a partition of $C_i$  for every $i=1,\ldots,r$.
  Suppose that  $C=C_i$ for some $i=1,\ldots,r$. If $i=r$, then $s(C)=\emptyset$. Then, we can assume that $i=1,\ldots,r-1$. Let $v\in s(C)$. By definition of the set $s(C)$, $N_{\sigma}(v)\not\subseteq C$. Therefore, there exists a vertex $z\in N_{\sigma}(v)$ such that $z\not\in C$. Since $\sigma^{-1}(v)<\sigma^{-1}(z)$ and $z\not\in C$, it follows that $z\in \bigcup\limits_{j=i+1}^{r}C_j$. Then, by construction of $\sigma$, we have that $\sigma^{-1}(u)<\sigma^{-1}(z)$ for every $u\in S_i=C\setminus s(C)$.

We proceed  by contradiction.   Suppose that there exits a vertex $u\in C\setminus s(C)$ with $\sigma^{-1}(v)<\sigma^{-1}(u)$. Then, $u\in N_{\sigma}(v)$. Since $N_{\sigma}(v)$ is a clique, we have that $uz\in E(G)$. As $z\not\in C$ and $N_{\sigma}(u)\subseteq C$, we have that $z\not\in N_{\sigma}(u)$. Since $uz\in E(G)$, we conclude that $u\in N_{\sigma}(z)$.
This means that $\sigma^{-1}(z)<\sigma^{-1}(u)$, a contradiction.

  \item [(c)] Suppose that $C=\{u_1,u_2,\ldots,u_{|C|}\}$ is well ordered. By (b) it follows that $u_j\not\in s(C)$ for every $j=1,\ldots,|C|-|s(C)|$. Since $N_{\sigma}(u_j)\subset C$ for every $j=1,\ldots,|C|-|s(C)|$, we have that $n_{\sigma}(u_j)=|C|-j$. Therefore, there exists a unique vertex $u\in C \setminus s(C)$ such that $n_{\sigma}=i-1$, because   $|s(C)|<i\le |C|$.

   \item [(d)] Let $v\in C\cap C'$ . We consider the following cases.

\noindent\underline{\emph{ Case (i):  $N_{\sigma}(v)\cap (C'\setminus C)\neq\emptyset$.}}
  In this case $N_{\sigma}(v)\cap (C\setminus C')=\emptyset$, because $N_{\sigma}(v)$ is a clique.
  We deduce that $\sigma^{-1}(u)<\sigma^{-1}(v)$ for every $u\in C\setminus C'$, since $u\not\in N_{\sigma}(v)$. Let $v'\in C\cap C'$ be any other vertex different from $v$. If $v'\in N_{\sigma}(v)$, then $\sigma^{-1}(u)<\sigma^{-1}(v)<\sigma^{-1}(v')$  for every $u\in C\setminus C'$. If $v'\not\in N_{\sigma}(v)$. As $N_{\sigma}(v)\subset N_{\sigma}(v')$, it follows that $N_{\sigma}(v')\cap (C'\setminus C)\neq\emptyset$. Then, by a similar argument applied to $v'$, we conclude that
 $\sigma^{-1}(u)<\sigma^{-1}(v')$ for every $u\in C\setminus C'$.

\noindent  \underline{\emph{Case (ii): $N_{\sigma}(v)\cap (C\setminus C')\neq\emptyset$.}}
The proof is analogous to Case (i).

\noindent\underline{\emph{Case (iii): $N_{\sigma}(v)\subset (C\cap C')$.}}
 In this case, $\sigma^{-1}(u)<\sigma^{-1}(v)$ for every $u\in C\cup C'\setminus (C\cap C')$.  Let $v'\in C\cap C'$ be any other vertex different from $v$. If $v'\in N_{\sigma}(v)$, then $\sigma^{-1}(u)<\sigma^{-1}(v)<\sigma^{-1}(v')$ for every $u\in C\cup C'\setminus (C\cap C')$. If $v'\not\in N_{\sigma}(v)$, we assume  that $N_{\sigma}(v')\cap (C'\setminus C)\neq\emptyset$. Then by a similar argument to Case (i), we have that $\sigma^{-1}(u)<\sigma^{-1}(v')$ for every $u\in C\setminus C'$. The case when $N_{\sigma}(v')\cap (C\setminus C')\neq\emptyset$ is analogous. The case $N_{\sigma}(v')\subset C\cap C'$ implies that $\sigma^{-1}(u)<\sigma^{-1}(v')$ for every $u\in C\cup C'\setminus(C\cap C')$.

We conclude, in any case, that $\sigma^{-1}(u)<\sigma^{-1}(v)$ for every $u\in C\setminus C'$ and every $v\in C\cap C'$ or   $\sigma^{-1}(u)<\sigma^{-1}(v)$ for every $u\in C'\setminus C$ and every $v\in C\cap C'$.
\end{itemize}
\end{proof}

\subsection{A combinatorial shifting of chordal graphs}
In this subsection, we define a combinatorial  procedure which takes chordal graphs to threshold graphs preserving the clique complex.
This procedure results  a shifting operation which allows us  to have a combinatorial description of shifting in order to use in our results
(for a different  study of combinatorial shifting of chordal graphs we refer to Murai's work on shifting for graphs \cite{Murai}).

\begin{definition}\label{combishiftingraph}
Let $G$ be a chordal graph with order $n$ and let $K_d=\{x_1,x_2\ldots,x_d\}$ be a maximum clique of $G$.
Let  $\sigma$ be a PEO  of $G$ that satisfies the conditions of Lemma \ref{K_d} for $G$ and $K_d$ (in particular $\sigma^{-1}(x_i)=n-i+1$ for every $i=1,\ldots,d$).
 For every
 vertex $u\in V(G)\setminus K_d$ suppose that $N_{\sigma}(u)=\{u_{1},u_{2},\ldots,u_{n_{\sigma}(u)}\}$ is well ordered.
 We define a map $\alpha_{\sigma}$  from $E(G)$ to the set of all $2$--subsets of $V(G)$ as follows:
\begin{itemize}
  \item  [(a)] $\alpha_{\sigma}(uv)=uv$ for every edge $uv\in E(K_d)$.
  \item  [(b)] $\alpha_{\sigma}(uu_{i})=ux_i$ for each vertex $u\in V(G)\setminus K_d$ and every $i=1,\ldots,n_{\sigma}(u)$.
\end{itemize}
We define  the graph $\alpha_{\sigma}(G):=(V(G),\alpha_{\sigma}(E(G)))$.
\end{definition}

Figure \ref{chordalthreshold} shows a chordal graph $G$ and the graph $\alpha_{\sigma}(G)=T$. The labeling in the figure corresponds to a PEO $\sigma$ of $G$ that satisfies the conditions of Lemma \ref{K_d}. Vertices $7,8,9,10$ corresponds to $K_d=\{x_1=\sigma(10),x_2=\sigma(9),x_3=\sigma(8),x_4=\sigma(7)\}$. For instance, we observe that if $u\in V(G)\setminus K_d$ is such that $u=\sigma(2)$, as $N_{\sigma}(u)=\{u_{1}=\sigma(3),u_{2}=\sigma(4)\}$ then $ux_1$ and $ux_2$ are edges in $T$. On the other hand, we may observe that the PEO $\sigma$ of $G$ in Figure \ref{chordalthreshold2}  also satisfies the conditions of Lemma \ref{K_d}.

\begin{figure}[htb]
\begin{center}
 \includegraphics[width=.7\textwidth]{ChordalThreshold.pdf}
\caption{ \label{chordalthreshold}}
\end{center}
\end{figure}

We now show that the previous graph shifting $\alpha_{\sigma}(G)$ yields the same $b$--vector of $G$. This is a key tool as it yield the same output as exterior or symmetric shifting. However, its construction allows to see that certain combinatorial properties are preserved.
\begin{theorem}\label{samecliquevector}
Let $G$ be a chordal graph and let $T=\alpha_\sigma(G)$ be as in Definition \ref{combishiftingraph}.
Then  $T$ is a threshold graph and both graphs have the same clique vector and $b$--vector.
\end{theorem}
\begin{proof}

 We note, by the definition of $\alpha_{\sigma}$,  that $T$ is conformed two parts. First, the maximum clique $K_d=\{x_1,x_2\ldots,x_d\}$,  where we may assume that $\sigma^{-1}(x_i)=n-i+1$ for every $i=1,\ldots,d$. Second,  additional vertices $v$ adjacent to $K_d$ such that if $v\in V(T)\setminus K_d$ with $\deg(v)=i$.
Then, $N(v)=\{x_1,x_2\ldots,x_i\}$.

 In order to prove that $T$ is threshold, it is enough by Theorem \ref{Mahadev} to show that $T$ does not contain graphs $C_4$, $2K_2$ or $P_4$ as an induced subgraph. It is not difficult to see that $T$ does not contain the graphs $C_4$ or $2K_2$ as an induced subgraph.  By means of contradiction, suppose that $T$  contains the graph $P_4=p_1,p_2,p_3,p_4$ as an induced subgraph. By the construction of $T$ we notice that $|P_4\cap K_d|\ge2$. If $|P_4\cap K_d|\ge3$ we would have a triangle in the induced subgraph $P_4$ which is a contradiction. Therefore, $|P_4\cap K_d|=2$, say
  $P_4\cap K_d=\{ p_2,p_3\}$. Then,  $p_1,p_4\not\in K_d$. With out loss of generality , we assume that $\deg(p_1)\le \deg(p_4)$.
Then, $N(p_1)\subseteq N(p_4)$  by  definition of the graph $T$. We conclude that $p_2p_4\in E(T)$, a contradiction. Similar arguments holds if another two vertices of $P_4$ are in $K_d$. Therefore  the graph $T$ is threshold.

We now show that $G$ and $T$ have the same clique vector, i.e,  $\mathbf{c}(G)=\mathbf{c}(T)$. We note that $|V(G)|=|V(T)|$, and so, $\mathbf{c}(G)$ and $\mathbf{c}(T)$ coincide in its first coordinate.
We now prove that $\alpha_{\sigma}$ induces a bijection between the set of $r$--cliques of $G$ and the set of $r$--cliques of $T$ for each $r=2,3,\ldots,d$.
 Let $C_u=\{u_1,u_2,\ldots,u_r\}$ be a well ordered $r$--clique of $G$ not contained in $K_d$.
Note that if $u_i\in K_d$ for some $i\in \{1,\ldots,r\}$. Then, $u_{j}\in K_d$ for every $j>i$, because $\sigma^{-1}(u_i)<\sigma^{-1}(u_j)$ and
 $\sigma^{-1}(z)<\sigma^{-1}(x)$ for every $z\in V(G)\setminus K_d$ and every $x\in K_d$. Hence, $u_1\not\in K_d$. Consider $N_{\sigma}(u_1)=\{u_1',u_2',\ldots,u_{n_{\sigma}(u_1)}'\}$, which is well ordered. As
 $\{u_2,u_3\ldots,u_r\}\subseteq N_{\sigma}(u_1)$, there exists $\{j_2,j_3,\ldots,j_r\}\subseteq \{1,\ldots,n_{\sigma}(u_1)\}$ such that $u_{j_q}'=u_q$ for every $q=2,3,\ldots,r$. Therefore, the map $\alpha_{\sigma}$ sends the clique $C_u$ in $G$ to a unique clique $\alpha_{\sigma}(C_u)=\{u_1,x_{j_2},x_{j_3},\ldots,x_{j_r}\}$ in $T$. Let $C_v$ be any $r$--clique  of $G$ such that  $\alpha_{\sigma}(C_u)=\alpha_{\sigma}(C_v)$.  We now show that $C_u=C_v$.
 We observe first that $C_v$ is not contained in $K_d$.
 Otherwise,  we would have that $\{u_1,x_{j_2},x_{j_3},\ldots,x_{j_r}\}\subseteq K_d$, because the map $\alpha_{\sigma}$ is the identity in $K_d$. This yields a contradiction because $u_1\not\in K_d$.

 Consider the well-ordered sets  $C_v=\{v_1,v_2\ldots,v_r\}$ and  $N_{\sigma}(v_1)=\{v_1',v_2',\ldots,v_{n_{\sigma}(v_1)}'\}$.  Then $v_1\not\in K_d$. Since
  $\{v_2,v_3\ldots,v_r\}\subseteq N_{\sigma}(v_1)$, there exists $\{i_2,i_3,\ldots,i_r\}\subseteq \{1,\ldots,n_{\sigma}(v_1)\}$ such that $v_{i_q}'=v_q$ for every $q=2,3,\ldots,r$. Therefore, the map $\alpha_{\sigma}$ sends the clique $C_v$ in $G$ to a unique clique $\alpha_{\sigma}(C_v)=\{v_1,x_{i_2},x_{i_3},\ldots,x_{i_r}\}$ in $T$.  As $\alpha_{\sigma}(C_u)=\alpha_{\sigma}(C_v)$, it follows that $u_1=v_1$. Since
 $\sigma$ is a bijection $\sigma:\{1,\ldots,n\}\to V(G)$, we conclude that $i_q=j_q$ for every $q=2,3,\ldots,r$. This means $C_u=C_v$.

 Finally,  we have that $\alpha_{\sigma}(C)\neq\alpha_{\sigma}(C')$ for any distinct $i$--cliques $C$ and $C'$, both contained in $K_d$. Therefore $\alpha_{\sigma}$ induces a bijection between the set of $r$--cliques of $G$ and the set of $r$--cliques of $T$ for each $r=2,3,\ldots,d$. We conclude that $\mathbf{c}(G)=\mathbf{c}(T)$. Hence, $\mathbf{b}(G)=\mathbf{b}(T)$.
\end{proof}

\begin{corollary}\label{corocombishift}
Let $G$ be a chordal graph and let $T=\alpha_\sigma(G)$ be as in Definition \ref{combishiftingraph}. Then  $\Delta(T)=\Delta(G)^e$. In particular, $\kappa(G)=\kappa(T)$.
\end{corollary}
\begin{proof}
By Theorems \ref{TeoCarly} and \ref{samecliquevector}, $\Delta(T)$ is shifted and $T$ is threshold respectively. We also have that $\Delta(G)^e$ is shifted. By Theorem \ref{TeoCarly}, there exists a threshold graph $T'$ such that  $\Delta(G)^e=\Delta(T')$. Then, $T'$ and $G$ have the same clique vector.   By Theorem \ref{samecliquevector}, $T$ and $G$ also have the same clique vector.
Since $T$ and $T'$ are threshold graphs with the same clique vector, they also have the same $b$--vector. Then, $T=T'$ and   $\Delta(T)=\Delta(G)^e$.
  Since exterior  shifting operation preserves depth by Theorem \ref{ThmShiftProp}, we
   have that $\Depth(R/I_{\Delta(G)})=\Depth(R/I_{\Delta(T)})$.
   Since $G$ and $T$ are chordal graphs, then $I_{\Delta(G)}$ and $I_{\Delta(T)}$ have $2$-linear resolutions \cite[Theorem 1]{Fro}. We conclude that
   $\kappa(G)=\kappa(T)$ by Corollary \ref{CorDepth}.
\end{proof}

\begin{notation}
Let us define $\widetilde{\kappa}(G):=\max\{|C\cap C'|\mbox{ }\vert \mbox{ }C  \mbox{ and } C' \mbox{ are maximal cliques of $G$ }\},$
\end{notation}

 We note that $\kappa(G)\le \widetilde{\kappa}(G)$.
Next, we  show  that $d_i(T)\le d_i(G)$ and the equality holds for $i> \widetilde{\kappa}(G)$.

\begin{theorem}\label{D_i}
Let $G$ be a chordal graph and let $T=\alpha_\sigma(G)$ as in Definition \ref{combishiftingraph}. Then,
\begin{itemize}
  \item [(a)] $d_i(T)\le d_i(G)$ for every $i=1,\ldots,d$.
    \item [(b)] $d_i(T)=d_i(G)$ for every $i> \widetilde{\kappa}(G)$.
\end{itemize}
\end{theorem}
\begin{proof}By Definition \ref{combishiftingraph},
  $K_d=\{x_1,x_2\ldots,x_d\}$ is a maximum clique of $G$ with $\sigma^{-1}(x_j)=n-j+1$ for every $j=1,\ldots,d$.
\begin{itemize}
\item[(a)] We note that the claim  holds for $i=1$.

 Let $\mathcal{D}_i(G)$ be a minimum dominating $i$--clique in $G$ for some $i\in \{2,3,\ldots,d\}$.
Let $C_i\in  \mathcal{D}_i(G)$. Suppose that there exits a maximal clique $C$ containing $C_i$ with $i>|s(C)|$ Then, by Lemma \ref{K_d}(c),  there exists a unique vertex,  $u_C$, in $C\setminus s(C)$ such that $n_{\sigma}(u_C)=i-1$.
We define a map $\varphi$ from $\mathcal{D}_i(G)$ to $\mathcal{D}_i(T)$ as follows. For $C_i\in \mathcal{D}_i(G)$, we set
\begin{itemize}
  \item [(i)] If for every maximal clique $C$ of $G$ containing $C_i$ we have $i\le |s(C)|$, we set $\varphi(C_i)=\{x_1,x_2,\ldots,x_i\}$.
    \item [(ii)]  If there exists a maximal clique $C$ containing $C_i$ such that $i>|s(C)|$, we set $\varphi(C_i)=\{x_1,x_2,\ldots,x_{i-1},u_C\}$.
\end{itemize}
First, we  show that $\varphi$ is a map from $\mathcal{D}_i(G)$ to $\mathcal{D}_i(T)$.
 Suppose that (ii) occurs and that there is a maximal clique $C'$, different from $C$, containing $C_i$ and such that $i>|s(C')|$. By Lemma \ref{K_d}(c),  there exists a unique vertex $u_{C'}\in C'\setminus s(C')$ such that $n_{\sigma}(u_{C'})=i-1$. We prove that $u_C=u_{C'}$. By Lemma \ref{K_d}(d), we may assume without loss of generality that $\sigma^{-1}(v)<\sigma^{-1}(u)$ for every $v\in C\setminus C'$ and every $u\in C\cap C'$. Then, by Lemma \ref{K_d}(b), we have that $s(C)\subset C\cap C'$. If $C=\{u_1,u_2,\ldots,u_{|C|}\}$ is well ordered, it follows that $u_j\in C\cap C'$ for every $j=|C|-|C\cap C'|+1,|C|-|C\cap C'|+2,\ldots,|C|$. Moreover, since $u_j\not\in s(C)$ for every $j=1,\ldots,|C|-|s(C)|$ by Lemma \ref{K_d}(b), it follows that
  $n_{\sigma}(u_j)=|C|-j$ for every $j=1,\ldots,|C|-|s(C)|$. Since $|C\cap C'|\ge i>
|s(C)|$, we have that $|C|-i+1\in \{1,\ldots,|C|-|s(C)|\}$. We conclude that $n_{\sigma}(u_{|C|-i+1})=i-1$. As $u_{|C|-i+1}\in C\cap C'$, it follows that $u_{|C|-i+1}=u_C=u_{C'}$, because $u_C$ and $u_{C'}$ are unique in $C$ and $C'$ respectively with the property   $n_{\sigma}(u_C)=n_{\sigma}(u_{C'})=i-1$. Therefore, $\varphi$ is a map from $\mathcal{D}_i(G)$ to $\mathcal{D}_i(T)$.

Second,  we  show that $\varphi$ is surjective proving that $d_i(T)\le d_i(G)$ for every $i=2,3,\ldots,d$. By Observation \ref{obs}(e), there is only one minimum dominating $i$--clique in $T$ which is formed by $\mathcal{C}_i(T)$, the set of maximal $i$--cliques of $T$, and the $i$--clique $\{x_1,x_2,\ldots,x_i\}$.
We have that $\{x_1,x_2,\ldots,x_i\}\in \varphi(\mathcal{D}_i(G))$, since there is an $i$--clique $C_i\in \mathcal{D}_i(G)$ such that $C_i\subseteq K_d$. We note that a maximal $i$--clique in $T$ is a clique $\{x_1,x_2,\ldots,x_{i-1},u\}$ where $u\in V(T)\setminus K_d$. We now show that $\{x_1,x_2,\ldots,x_{i-1},u\}\in \varphi(\mathcal{D}_i(G))$.
By the definition of $\alpha_{\sigma}$, we have that $n_{\sigma}(u)=i-1$. We consider a maximal clique $C$ containing
 the $i$--clique $u\cup N_{\sigma}(u)$ in $G$, which means that $u\not\in s(C)$. Thus, it follows   that $s(C)\subset N_{\sigma}(u)$ by Lemma \ref{K_d}(b). As  $|N_{\sigma}(u)|=i-1$, we have that $i>|s(C)|$. We conclude that  $\varphi(u\cup N_{\sigma}(u))=\{x_1,x_2,\ldots,x_{i-1},u\}$, which proves that  $\varphi$ is surjective. Therefore $d_i(T)\le d_i(G)$ for every $i=1,\ldots,d$.

 \item[(b)] Suppose that $\mathcal{D}_i(G)=\{C_1,C_2,\ldots, C_{d_i(G)}\}$.  Let $M_j$ be a maximal clique of $G$ containing $C_j$ for every $j=1,\ldots,d_i(G)$.
  We have that
 $C_j$ nor $C_r$ are contained in $M_j\cap M_r$ for every distinct $j,r=1,\ldots,d_i(G)$, because $i>\widetilde{\kappa}(G)$.
 As a consequence, $M_j\neq M_r$. By definition of $s(M_j)$, we have that $\widetilde{\kappa}(G)\ge s(M_j)$ concluding that  $i>s(M_j)$. Then, by Lemma \ref{K_d}(c), there exists a unique vertex $u_j\in M_j\setminus s(M_j)$ such that $n_{\sigma}(u_j)=i-1$ for every $j=1,\ldots,d_i(G)$.

We  prove that the map $\varphi$ is injective for $i> \widetilde{\kappa}(G)$. By definition of  $\varphi$, we have that $\varphi(C_j)=\{x_1,x_2,\ldots,x_{i-1},u_j\}$ and $\varphi(C_r)=\{x_1,x_2,\ldots,x_{i-1},u_r\}$ for every distinct $j,r=1,\ldots,d_i(G)$.
We show that $u_j\neq u_r$ proving that $\varphi(C_j)\neq \varphi(C_r)$. We have $u_j\neq u_r$ if $M_j\cap M_r=\emptyset$.
 Suppose $M_j\cap M_r\neq\emptyset$. As $M_j\neq M_r$, we may suppose without loss of generality   that $\sigma^{-1}(u)<\sigma^{-1}(v)$ for every $u\in M_j\setminus M_r$ and every $v\in M_j\cap M_r$ by  Lemma \ref{K_d}(d). Since $i>|M_j\cap M_r|$ and   $n_{\sigma}(u_j)=i-1$,
 it follows that $u_j\in M_j\setminus M_r$. Then, $u_j\neq u_r$, and so, $\varphi$ is injective. Hence, $d_i(G)=d_i(T)$ for every $i> \widetilde{\kappa}(G)$.
 \end{itemize}
\end{proof}

\subsection{$b$--vectors and betti numbers of chordal graphs}

In this subsection we show our main result regarding the $b$--vector of a chordal graph.
In particular, we compute some of the entries of the $b$--vector. We first compute some entries using the combinatorial procedure $\alpha_\sigma$ introduced
in the previous subsection.

\begin{corollary}\label{b_i}
  Let $G$ be a chordal graph, then
  \begin{itemize}
    \item [(a)] $b_i\le d_i(G)$ for every $i=1,\ldots,d$;
    \item [(b)] $b_i=d_i(G)$ for every $i>\widetilde{\kappa}(G)$;
    \item [(c)] $b_i\le b_j$  for every $i,j>\widetilde{\kappa}(G)$ with $j<i$.
  \end{itemize}
\end{corollary}
\begin{proof}
By Theorem \ref{D_i} and Proposition  \ref{dominate}(c), we have (a) and (b). Let $\mathcal{C}_{\ge i}$ be the set of all maximal cliques of $G$ with cardinality at least $i$ and notice that $|\mathcal{C}_{\ge i}|=d_i(G)$ when  $i\ge\widetilde{\kappa}(G)+1$. Hence by (b) we have that $|\mathcal{C}_{\ge i}|=b_i$ for every  $i\ge\widetilde{\kappa}(G)+1$. As  $|\mathcal{C}_{\ge i}|\le|\mathcal{C}_{\ge j}|$ for every $j<i$, we have (c).
\end{proof}
The next example shows that  Theorem \ref{D_i}(b) and Corollary \ref{b_i}(b)and (c) are best possible.
\begin{example}\label{bestpossible}
For every positive integers $\kappa,\widetilde{\kappa}$ with $\kappa\le \widetilde{\kappa}$, consider the graph $G$ conformed by a clique $K=\{x_1,x_2,\ldots,x_{2\widetilde{\kappa}}\}$ of size $2\widetilde{\kappa}$ together with three additional  vertices, $u_{\kappa},u_{\widetilde{\kappa}}$ and $v_{\widetilde{\kappa}}$ such that $N(u_{\kappa})=\{x_1,x_2,\ldots,x_{\kappa}\}$, $N(u_{\widetilde{\kappa}})=\{x_1,x_2,\ldots,x_{\widetilde{\kappa}}\}$ and $N(v_j)=\{x_{\widetilde{\kappa}+1},x_{\widetilde{\kappa}+2},\ldots,x_{2\widetilde{\kappa}}\}$.

We observe that the graph $G$ of Example \ref{bestpossible} is chordal, $\kappa=\kappa(G)$ and $\widetilde{\kappa}=\widetilde{\kappa}(G)$. In addition,
 $b_i=d_i(T)$ for every $i=1,\ldots,d$ where $T$ is the threshold graph such that $\alpha_{\sigma}(G)=T$  by  Proposition \ref{dominate}(c) and Theorem \ref{samecliquevector}. Now, let us consider $i\le \widetilde{\kappa}$. We first notice that $b_i=2<3=d_i(G)$ for  $i=\kappa+1$ (which means $\kappa<\widetilde{\kappa}$ since $i\le \widetilde{\kappa}$). Second, we notice that  $b_i=1<2=d_i(G)$ for $i\neq\kappa+1$. In any case $b_i<d_i(G)$  if $i\le \widetilde{\kappa}$, showing that   Theorem \ref{D_i}(b) and Corollary \ref{b_i}(b) are best possible. Finally, if $i=\kappa=\widetilde{\kappa}$, we have that
$b_i=1<4=b_{i+1}$ showing that  Corollary \ref{b_i}(c) is best possible.
\end{example}

We now study the relation of the $b$--vector of a chordal graph $G$ with  the Betti numbers of the square-free monomial
ideal associated to the clique simplicial complex $\Delta(G)$. We now show that the Betti numbers of $R/I_{\Delta(G)}$ can be expressed by a formula in terms of  its $b$--vector when $G$ is a chordal graph.
\begin{remark}\label{relacion c vs b}
The relation between the $c$--vector and the $b$--vector (given indirectly by  Formula \ref{eq1}) can be written explicitly via the formula $c_i=\sum\limits_{j=i}^{d}\binom{j-1}{j-i}b_j$.
\end{remark}

\begin{proposition}\label{beta y b}
Let $G$ be a chordal graph,
 then the Betti numbers of $R/I_{\Delta(G)}$ can be expressed by a formula in terms of  its $b$--vector.
\end{proposition}
\begin{proof}
Since the $f$--vector and the clique vector of $\Delta(G)$ have the relation  $f_{i-1}=c_{i}$ for every $i=1,\ldots,d$ by Remarks \ref{h-vector f-vector} and \ref{relacion c vs b},  the $h$--vector and the $b$--vector of $\Delta(G)$ have the following relation
\begin{equation}\label{h y b}
h_i=\sum\limits_{j=0}^{i}(-1)^{i-j}\binom{d-j}{i-j}\Bigg( \sum\limits_{k=j}^{d}\binom{k-1}{k-j}b_k\Bigg)
\end{equation}
for every $i=0,1,\ldots,d$. Since $G$ is chordal, $I_{\Delta(G)}$ has a $2$-linear resolution \cite[Theorem 1]{Fro},
 we have that
 \begin{equation}\label{beta y h}
 \beta_i(R/I_{\Delta(G)})=\sum\limits_{\ell=0}^{2+i}(-1)^{\ell+i+1}h_{2+i-\ell}\binom{n-d}{\ell}
 \end{equation}
 for every $0\le i\le p$  by Theorem \ref{ThmBettiH}. Here, $p$ denotes the projective dimension of the Stanley-Reisner ring $R$. Therefore, by combining Equations (\ref{h y b}) and (\ref{beta y h}), we  obtain a formula of the Betti numbers $\beta_i(R/I_{\Delta(G)})$ in terms of its $b$--vector. Precisely $$\beta_i(R/I_{\Delta(G)})=\sum\limits_{\ell=0}^{2+i}(-1)^{\ell+i+1}\Bigg(\sum\limits_{j=0}^{2+i-\ell}(-1)^{2+i-\ell-j}\binom{d-j}{2+i-\ell-j}
\Bigg(\sum\limits_{k=j}^{d}\binom{k-1}{k-j}b_k\Bigg)
\Bigg)\binom{n-d}{\ell}$$  for every $0\le i\le p$.
\end{proof}

We now extend a theorem of Goodarzi's \cite[Theorem 1]{Goodarzi} regarding $b$--vector and connectivity of $G$. In particular, we describe one more entry in terms of connectivity components.

\begin{theorem}\label{k+1}
  Let $G$ be a chordal graph of order $n$ and vertex connectivity $\kappa$.
Then, $$b_{i}=
\sum\limits_{|Y|=i-1} (W(G-Y)-1)+1=\beta_{n-i}(R/I_{\Delta(G)})+1$$ for every $i=1,\ldots,\kappa+1$ where $Y\subseteq V(G)$. In addition,
$$b_{i}<
\sum\limits_{|Y|=i-1} (W(G-Y)-1)+1=\beta_{n-i}(R/I_{\Delta(G)})+1$$ for every $i=\kappa+2,\ldots, d$.
\end{theorem}
\begin{proof}
By Proposition \ref{beta}, we have that
$$
\beta_{n-i,n-i+1}(R/I_{\Delta(G)})=\sum_{|Y|=i-1} (W(G-Y)-1)+1=0
$$
for every $i=1,\ldots,\kappa$. Since we already know that $b_1=\ldots=b_\kappa=1$ from Goodarzi's work \cite[Theorem 1]{Goodarzi}, the theorem holds for every $i=1,\ldots,\kappa$.

We now focus for the case $i=\kappa+1.$
Let $T=\alpha_\sigma(G)$ as in Definition \ref{combishiftingraph}, then   $\Delta(T)=\Delta(G)^e$ and $\kappa(T)=\kappa(G)$ by Corollary \ref{corocombishift}.
We have that $b_{\kappa+1}(G) =b_{\kappa+1}(T)$ by Theorem \ref{samecliquevector}. In addition,
 \begin{align*}
\sum_{|Y|=\kappa} (W(G-Y)-1)
&= \beta_{n-i,n-i+1}(R/I_{\Delta(G)}) \hbox{ by Proposition \ref{beta}}\\
&=\beta_{n-i}(R/I_{\Delta(G)}) \hbox{ because }I_{\Delta(G)}\hbox{ has a }2\hbox{-linear resolution.}\\
&=\beta_{n-i}(R/I_{\Delta(T)}) \hbox{ by Proposition  \ref{PropPreservationBetta}}\\
& =\beta_{n-i,n-i+1}(R/I_{\Delta(T)}) \hbox{ because }I_{\Delta(G)}\hbox{ has a }2\hbox{-linear resolution.}\\
&=\sum_{|Y|=\kappa} (W(G-Y)-1) \hbox{ by Proposition \ref{beta}}.
\end{align*}
Then, it suffices to prove our claims for $T$, which were proved in Propositions \ref{dominate}(d) and \ref{k+2} respectively.

\end{proof}

The $b$--vector
 of $G$ in Figure \ref{chordalthreshold} is $(1,4,3,2)$ and $\kappa(T)=\kappa(G)=\widetilde{\kappa}(T)=\widetilde{\kappa}(G)=1$. We observe that effectively $b_1=1$ and $b_i=d_i(G)$ for every $i\ge 2$ (Corollary \ref{b_i}(b)). Figure \ref{chordalthreshold2} shows another chordal graph $G$ and its graph $\alpha_{\sigma}(G)=T$. We observe that the $b$--vector of $G$ in Figure \ref{chordalthreshold2} is $(1,2,3,2,2)$ and $\kappa(T)=\kappa(G)=1<2=\widetilde{\kappa}(G)<\widetilde{\kappa}(T)=4$. Hence, $b_1=1$ and $b_i=d_i(G)$ for every $i>2=\widetilde{\kappa}(G)$ (Corollary \ref{b_i}(b)). Finally, we observe that edges $(4,10)$ and $(9,10)$ of $T$ are a minimum  dominating $2$--clique in $T$ and edges $(3,5)$, $(4,5)$ and $(6,7)$ are a minimum  dominating $2$--clique in $G$. Hence, $b_2=d_2(T)=2<3=d_2(G)$. Nevertheless, we know that $b_2$ give us another information in  $G$, because  $b_2=\sum\limits_{|Y|=1} (W(G-Y)-1)+1=2$  by Theorem \ref{k+1} .
\begin{figure}[htb]
\begin{center}
 \includegraphics[width=.9\textwidth]{chordalthreshold3.pdf}
\caption{ \label{chordalthreshold2}}
\end{center}
\end{figure}

\subsection{Examples}
We finish this section by providing examples of pure and matroid complexes arising from chordal and threshold graphs in which we know the value of its $b$--vector.
A simplicial complex $\Delta$ is \emph{pure} if all its facets have the same cardinality and a \emph{matroid complex} $\Delta_V$ on a set of vertices $V$ is a pure simplicial complex such that $\Delta_{V\setminus S}$ is pure for every $S\subseteq V$.

Klivans \cite{Carly}  proved that given a graph $T$, $\Delta(T)$ is a shifted  matroid complex if and only if $T$ is threshold and its corresponding word is of the form $SDDDSSS$. The first example shows  that the above is true for pure complexes arising from threshold graphs.

\begin{example} \label{examplethresholdpure}
Let $T$ be a threshold graph
 and suppose that $\Delta (T)$ is pure.
Note that there is no maximal clique of size $i$ in $T$ for $i\le d-1$. Then, by Proposition  \ref{dominate}(i)(ii), $b_i=1$ for $i\le d-1$.  Since $\Delta(T)$ is pure the word corresponding to $T$ is of the form $SDDDSSS$,  $T-S$ is pure for every $S\subset V(G)$. This is because the corresponding word of $T-S$ is has the same type. Hence, $\Delta (T)$ is a matroid complex.
\end{example}

\begin{example}
Let $G$ be a chordal graph
 and suppose that  $\Delta(T)$ is a matroid complex.
 We show that $G$ is in fact threshold. For this, it is enough to show that $G$ does not contain the graphs $C_4$, $2K_2$ or $P_4$ as an induced subgraph  by Theorem \ref{Mahadev}.
First, we claim that for every vertex $v\in V(G)$ and every edge $u_1u_2\in E(G)$, we have that $vu_1\in E(G)$ or $vu_2\in E(G)$.
Let $v,u_1,u_2$ different vertices in $G$ such that $u_1u_2\in E(G)$ and let $S=V(G)-\{v,u_1,u_2\}$. As $\Delta (G)$ is a matroid complex, we have that $\Delta (G-S)$ is pure which means that $vu_1\in E(G)$ or $vu_2\in E(G)$.
Second, by the previous claim, $G$ does not contain the graphs $C_4$, $2K_2$ or $P_4$ as an induced subgraph. We obtain that  $G$ is threshold. Finally, we also notice that  $b_i=1$ for $i\le d-1$ by Example \ref{examplethresholdpure}.
\end{example}

The last example is when $G$ is a chordal graph and $\Delta(G)$ is pure. In this case,  the $b$--vector satisfies  $0<b_1\le b_2\le\cdots \le b_d $  \cite[Theorem 1.2]{H08}. Hence, from Corollary \ref{b_i}(c), we obtain the following.

\begin{example}
Let $G$ be a chordal graph with
 $\Delta(G)$ pure. Then,  $b_{\widetilde{\kappa}(G)+1}=b_{\widetilde{\kappa}(G)+2}=\cdots=b_d$.
\end{example}

\bibliographystyle{alpha}

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The Assad regime was found responsible for killing US journalist Marie Colvin in 2012, a US court has ruled. Vice-President of the Syrian Coalition Dima Moussa described the ruling as "very important." She called for punishing all crimes committed by the Assad regime against civilians in Syria using all available means, including law courts in which lawsuits can be brought against the Assad regime to hold its symbols accountable for the crimes it committed against the Syrian people.
The US district court for the District of Columbia ruled that the Assad regime was liable to pay $302m in damages to Colvin's family who filed a lawsuit against the regime for its responsibility for her death when regime forces shelled a media center in the neighborhood of Baba Amr in Homs in February 2012.
US District Court Judge Jackson ruled that the Assad regime was guilty of the extrajudicial killing of an American citizen by planning and carrying out an attack on a media center in the neighborhood of Baba Amr.
Prosecutors have proven through sufficient evidence that the Assad regime bears responsibility for Colvin's death. A claim against the Assad regime was lodged with a Washington court by her sister Cathleen and her three children.
Moussa stressed that the Assad regime bears primary responsibility for the murder of Colvin along with hundreds of journalists and hundreds of thousands of Syrian civilians. She pointed to the need to activate the International Impartial and Independent Mechanism (IIIM) which established in 2016 to support efforts to hold accountable perpetrators of war crimes in Syria.
Moussa also underscored the need to establish a competent tribunal court to investigate war crimes and crimes against humanity committed in Syria, especially in light of Russia's blocking of efforts on the Security UN Council to refer Syria to the International Criminal Court.
Moreover, Moussa stressed that doing justice in Syria is key to finding a just, lasting political solution in the country in accordance with the relevant international resolutions, most importantly the Geneva Communique of 2012 and UN Security Council Resolution 2254.
Colvin was killed in sustained artillery barrages by the Assad regime on the apartment building housing the media center in Bab Amr along with French photographer Remi Ochlik. British photographer Paul Conroy, who was working with Colvin, was injured along with Syrian media activists before Assad forces stormed the area.


Golden Globe®-nominated Empire continues to be one of the most popular and acclaimed shows on television. A powerful drama about a family dynasty set within the glamorous and sometimes dangerous world of hip-hop, the show revolves around the Lyons and their media company, Empire Entertainment.
A group of ex-Army doctors work the night shift at a hospital and do whatever it takes to help their patients. Using battlefield-tested medical skills, doctors clash with hospital bureaucrats who are more worried about the bottom line than the lives of their patients.
Starring: Eoin Macken, Jill Flint, Scott Wolf, Brendan Fehr, Robert Bailey Jr.
Following the execution of Anne Boleyn, King Henry VIII remarries in hopes of having a son as challenges to his authority from inside and outside the kingdom threaten his hold on England's throne.
Set in the '90s, hip-hop loving teenager Eddie Huang and his family have lived in Orlando for a few years now and have assimilated nicely into the suburban American lifestyle. Cultural differences still present everyday challenges, but close friends, neighbors and business acquaintances are there to help them navigate the complexities of raising a family of future millennials.
Before he was an actor, Ken Jeong was an honest-to-goodness M.D. Now as Dr. Ken, Jeong brings medicine and laughter together as a grumpy and beleaguered doctor whose challenging professional life is outdone only by his chaotic home life.
A former lawyer attends a community college when it is discovered he faked his Bachelor's degree. In an attempt to get with a student in his Spanish class, he forms a Spanish study group. To his surprise, more people attend the study group and the group of misfits form an unlikely community.


February 1897: ACHING ARM TO MAKE PANCAKES.
At a cookery-school I attended, we had a whole lesson devoted to pancakes. We separated the yolks of the eggs from the whites, mixed the yolks with the flour and then added milk to form a batter. Then we beat the whites to a stiff froth till our arms ached (mine did at any rate); this was to be stirred in gently that the pancake might be very light.
The joke was (as I rubbed my aching arm) that the pancakes were not nearly so good as those I make hastily at home, with no switching of eggs at all. Mix a pinch of salt and of baking powder with the flour, and with skim milk form a batter that will run easily from the spoon; pour the mixture into a jug.
Put a piece of dripping into the frying-pan and allow it to get smoking hot; then pour in sufficient batter to just cover the pan. When the surface is dry, sprinkle it well with sugar; turn over one of the edges with a knife and roll the pancake over and over until only the brown under-surface is visible. The pancake is now finished. As it takes four to five minutes for each, it requires some time to make sufficient for a large family.
This entry was posted in Cooking and tagged food, pancakes. Bookmark the permalink.
2 Responses to February 1897: ACHING ARM TO MAKE PANCAKES.
My Scotish grandparents ate pancakes this way…and I do, too!!! I have never heard of anyone else doing so!


text:
Q: Use javascript to change the href of a button depending on screensize I made a fluid grid based website, and now i want to add movie content within an iframe. Since the iframe is changeing size, i made the content i want to add in these 3 sizes:
-option1 size: 720p: 1280 x 720
-option2 size: 480p: 854 x 480
-option3 size: 360p: 640 x 360
now i'm looking for a script to measure the screen of the user, and then to change the href of the button to option 1, 2 or 3.
This is how far i come. i hope someone has an example.

            <img src="images/3D.png"

thanks in advance.

A: Another option. Use media queries and show/hide buttons based on the size.
CSS Media Way
<!DOCTYPE html>

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JavaScript way you were doing:
<!DOCTYPE html>

    LINKY

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text:
Popular Science Monthly/Volume 22/February 1883/Correspondence
< Popular Science Monthly | Volume 22 | February 1883
Sketch of Henri Milne-Edwards
Popular Science Monthly Volume 22 February 1883 (1883)
Editor's Table
637248Popular Science Monthly Volume 22 February 18831883
THE BICYCLE CONTROVERSY IN STOCKBRIDGE.
Messrs. Editors:
ALTHOUGH the writer belongs to those inhabitants of the village of Stockbridge who are editorially stigmatized in the December number of your magazine as destitute of "moral backbone," as "openly immoral," and "barbarians," yet cowards and criminals have their rights at the bar of editorial as of other justice, and he asks you to permit him to file a plea to your indictment—in other words, to publish this answer to your strictures.
Sound criticism, quite as much as sound philosophy, you will agree, depends on a correct and complete statement of the facts. The following version of the bicycle controversy can be maintained by many witnesses. The writer asserts it to be in every material part substantially true.
The professor, whom you justly call distinguished, fresh from a victory in the Central Park of this city over the ladies, invalids, and children who had been accustomed to be pushed about the skating-rink on the sliding-chairs, which doubtless interfered with his essays in skating, and perhaps on abstruser matters, came to his country retreat in Stockbridge naturally confident of equal success in clearing out all bicycles from his path. He was accustomed to walk with his eyes downcast, medians nescio quid nugarum philosophicarum totus in ilis, and the necessity of keeping wide awake with his bodily eyes was annoying. There was no Park Commissioner and no "Century" or other social club where officials can be button-holed, and petitions granted inter pocula; but there were three selectmen, chosen to guard the interests of the town in the old fashioned New England way. He drew a petition, procured eighteen signatures, and presented it to them. Taking it as a fair indication of the sentiment of the town, the selectmen were on the point of granting it, when the application chanced to become known to one or two inhabitants who took a different view. A remonstrance was drawn, based on the facts that accidents more commonly occurred from bicycles frightening horses in the roadway than from permitting them on sidewalks; on the hardship of practically depriving children of all use of the bicycles; on the impolicy and injustice of subjecting summer fugitives from the cities to the same kind of restrictions they had fled from; and denying the existence or seriousness of the so-called accidents alleged in the petition. This remonstrance was signed by thirty-six persons. The signers were generally heads of families, and up to the moment of this act of turpitude were (with the exception of the present writer) persons of recognized standing and character. In consequence of this remonstrance, the selectmen decided to ignore the petition. The shrewd professor, then perceiving that though, by dint of that persistent persuasiveness in which he is unexcelled, he might collect signatures, yet that all the names remaining in the town could not outweigh the remonstrance, called on the person who prepared that paper and urged his assent to a second petition, which had already been signed by a considerable number of visitors and residents. This was a remarkable document. It began by renewing the prayer of the original petition; but the various signers had been permitted to incorporate in it their different views and prejudices, which gave it so motley an aspect that it was hard to determine what was the net application. It asked in one place that all bicycles be excluded from the sidewalks; in another, that they be so excluded excepting children's; in another, that bicycles exceeding thirty-six inches in diameter be so excluded; in another, that bells be required to be attached to all bicycles; and, in another, that no bicycle be allowed to go anywhere in the village faster than five miles an hour. The person of whom the request was made, though reiterating his opinion that the real and only serious danger of accidents in the village was from the frightening of horses in the road, yet being fond of peace, something of a "moral coward," and willing to see how a compromise would work, yielded to the professor's strenuous demand, and reluctantly signed a memorandum to the effect that he considered that the chief objection to an order excluding bicycles from the sidewalks would be removed by permitting children to ride them there. The multiform document, thus re-enforced, was thereupon taken to the other signers of the remonstrance, who, seeing the memorandum of their representative, signed also.
The wheels of the opposition being thus scotched, other signatures were then obtained, to the number no doubt correctly stated by your informant as a hundred and sixty-eight. Meantime, a moment's cool reflection out of the range of the professor's battery having convinced the first compromiser that he had made a mistake, he sent a note to the professor, requesting that his name might be removed. When the note was received at the professor's house, he was out, busily engaged in getting signatures. The next day he called on the repentant signer and informed him that, before the note was received, the document had been already presented to the selectmen.
These selectmen, though I maintain just and well selected, were puzzled. After discussing in open session the question whether a thirty-eight-inch bicycle ridden by a small boy should be compelled to take to the road, while big boys on thirty-six-inch bicycles were allowed to patrol the sidewalks, and the difficulties of enforcing regulations depending on the exact size of the bicycle, they decided, as the wisest practical course, to exclude all bicycles, in accordance with the original, principal, and most definite purpose of the petition. This decision, we believe, was a perfectly honest one. There was, at any rate, enough to justify it as such, and the burden is on those who charge dishonesty to prove it. It is true, as alleged, that this order necessarily "caused irritation." Why? Because it was unjust, or appeared so to the village. Yet this indiscriminate exclusion was the real object at which the professor aimed, and which he partly gave up only in order to make sure of as much restriction as he could get.
But the order was not generally satisfactory, and a counter-petition (which your informant seems to have forgotten) was drawn up and at once signed by about two hundred of those who favored the cause of the bicycles. The list included a great majority of the substantial names of the village, and was handed in long before anything like a complete canvass could be made, because the chairman of the Board of Selectmen was to leave town the next morning for a considerable absence, and immediate action was necessary. The selectmen held another session. Horses were frightened daily under the eyes of all by bicycles on the road. No accidents known to the community at large had resulted from the bicycles on the sidewalks. The decision of the magistrates was a responsible one in a pecuniary sense, as they might entail on the town heavy damages as the result of forcing the bicycles into the road, to the distress and dismay of the riding and driving public, Still, prompted by the desire which they throughout showed to act cautiously and fairly on a controverted point with which they professed no personal acquaintance, they made, in accordance with the general sentiment, a modified and, as the event seemed to prove, a judicious order. They excluded the bicycles within certain designated limits from the central and business part of the village, where alone they thought accidents likely to occur, but permitted them elsewhere on the sidewalk. From extreme caution they proceeded tentatively, however. The order was made about the middle of August, to remain in force till the 1st of September. This was for the purpose of testing its practical operation before establishing a rule at the expiration of the time limited. No accident happened. Nothing happened which the lenses of the learned professor could magnify into an "accident"; and on the 1st of September the order was indefinitely continued, with the like beneficent result.
Now, the selectmen are charged with "playing into the hands of an active party in favor of the boys." We will not stop to inquire who was this active part} 7 which overawed the selectmen and commanded a majority of the votes. But is not it hard on the selectmen, who did what they conceived their duty in the manner described above, to charge them with knavery, without, so far as appears, other grounds than that the order they finally made, in view of all the facts, was unsatisfactory to the professor? This is not Spencerian doctrine. If it is what the editor of this magazine calls a "vigorous canon of scientific method," it is rather too vigorous.
In the article in the "Springfield Republican," the accusations (according to the writer's recollection) are more bitter, and one of the magistrates is attacked by name. We shall come presently to the charge of Stockbridge special amenableness to Mr. Spencer's indictment against the American people. But here we may inquire whether there is no other fault found by him with our people, to which those are amenable who publish accusations against official persons for which there is no scintilla of proof. Mr. Spencer's ethics prohibit slander and personalities.
Now let us glance at the more general facts:
It will be admitted that bicyclists, like other domestic animals, have some rights which, once defined, are as much entitled to protection as the wider liberty allowed pedestrians. The question to decide is, at what point the exercise of these rights begins to turn into a trespass. It will hardly be denied that in the open country bicyclists may leave the rough road for the more traversable path which runs by it, and when the road and the path lead into a small, straggling village, where pedestrians are rare, the practice is still clearly permissible. When the road and path lead into a larger village, the road becomes paved or macadamized, and the path becomes a sidewalk, the question changes its aspect. In villages like Stockbridge, where there is plenty of space, and the broad, graveled sidewalks are never crowded, it is not a simple one. A good bicyclist can guide his machine almost around a blade of grass, and the great preponderance of accidents have resulted from bicycles encountering horses in the road. In large towns with thronged sidewalks, it is obvious that, whether bicycles are to be allowed in the road or not, they are in the way on the sidewalk, and ought not to be permitted there. A rule adapted to the compact English villages is not necessarily the best guide for ours. But it is not true that bicycles are not permitted on their sidewalks by any such general rule as the professor states. There, as here, the rule depends on the size of the village, and on the question whether such permission would or not be practically inconvenient. When the controversy in Stockbridge was going on, and rival posters were daily going up, considerable evidence was collected to this effect. In some villages, as in "Henley on-Thames," the postmen travel altogether on bicycles, going freely on the sidewalks. In Massachusetts towns of about the size of Stockbridge, such as the neighboring villages of Lee and Barrington (which, by-the-way, are considerably larger than Stockbridge), and where there are no college professors, or, if there are, they reserve their perambulatory meditations for their return to their Poccile, bicycles are generally permitted on the sidewalks. In much more populous towns, like Pittsfield and Springfield, they arc doubtless excluded. Now, who is to decide when the permission to go on the sidewalk becomes a nuisance? Certainly, in this country, the people themselves, through their authorities. In New England these are the selectmen. New England almost if not quite alone retains the system of town self-government which for so many centuries preserved liberty and fostered civilization in Europe. The selectmen of a Berkshire village, charged with the supervision of the lives and property of the inhabitants, certainly have better means of judging and a stronger motive to judge rightly in what parts and places bicycles should be permitted within their jurisdiction, than a newspaper in Hampden County, or a magazine in New York City, acting on the ex parte statements of even a distinguished individual, who, having undertaken to have the question settled to suit himself, is indignant because the selectmen, in accordance with the expressed wish of a majority of those interested, have decided differently.
The writer is one of the professor's many admirers, and would be the last to impugn his good faith; but, once enlisted in a campaign, whether in behalf of science, charity, or some idiosyncrasy, he goes on with a persistency which is always indomitable and sometimes headlong; and he is apt to assume that his cause has but one side, which, in cases coming under the third of the above categories, at least, is not invariably true. In this instance, he threw aside the sketch-book with which it had been his wont to exercise his charming artistic talent during his vacation, and devoted his time to scouring and ransacking the by-ways and corners of the village for rumors and reports of accidents from bicycles, the existence of which, before they were seen or heard of by any one else, he was as satisfied of as Leverrier was of the existence of the planet Neptune before he saw it. He posted up from day to day notices of incidents more or less founded on fact, which he apotheosized into catastrophes. He went back over a period of four years, to the first introduction of the bicycle into the neighborhood, and the only genuine "accident" worth mentioning, it is believed, if his collection were dispassionately interpreted, was the result of the fall, on a baby carriage, of a bicycle unskillfully mounted by a learner at this early day. People saw with wonder lists of tragedies posted in front of the post-office which no one had heard of, and which were the more mysterious as they were generally without date, and the sufferers were commonly designated as Mr., Mrs., or Miss Blank, or the infant child of Mr. and Mrs. Blank. Why had no one heard of these terrible things? Why were not half of us in mourning or in tears? What right had we to walk erect on unbroken legs while so many were mangled? It seemed as if a wholesale slaughter had been going on in the midst of us without our knowledge. Fortunately, a little inquiry into the facts soon dissipated our apprehensions.
Let me illustrate: the writer, while looking at the list of casualties with feelings fortunately for him not quite akin to those with which we used to devour the returns from the Chickahominy, read somewhat as follows (the number and the precise words are not remembered, but the substance of the bulletins is correctly given): "No. 16. The infant child of Mrs. Blank run over and badly injured by a bicycle. For particulars refer to J. 0. R., Esq." Looking further the writer read as follows: "No. 21, J. O. R., Esq., run into and hurt by a bicycle." (J. O. R. was the same person referred to in No. 16.) It so happened that the writer observed at the moment in the post-office J. O. R., Esq., himself, and improved this opportunity by inquiring of him the particulars of the two calamities, calling his attention to the alarming record before us. "Well," said he at last, "that isn't exactly right. I didn't see anything happen to any child, and I haven't been run into myself, but I understood that old Mr. J. G. had been run into and hurt." Soon after this conversation, the writer chanced to meet the wife of the old gentleman in question, and said to her he was very sorry to hear that her husband had been run into by a bicycle, and hoped it was nothing serious. "Why, no," said she, "he was not actually run into, but you know he is quite infirm, and a bicycle came very near him and he was a good deal alarmed lest it might hit him." This was the residuum from the analysis of the two accidents. A list of burlesque "escapes" was posted from day to day by the side of that of the so-called "accidents" and about equal credence was given to each. Both were worthy of "Patience," or the "Belle Hélène." To the annoyance of both parties, the selectmen at last impartially took down both sets of notices, not because of any improper motives as charged, but because of the crowd which gathered daily in front of the post-office so thickly as to become a nuisance. The professor was laughed out of court; and we villagers, while all conceding his excellence and genius, think him daft on the bicycle question. The real bullying was all done by him, though unconsciously; and the real "cowards" were those who, like the writer, under the glare of those wonderful eyes, had not dared to refuse to sign his pieced and patched compromise petition.
In sooth, this particular charge of moral cowardice in the Stockbridge people is unfounded and ridiculous. Not only were there no accidents deserving the name, but none were likely to occur on the sidewalk, which might not be avoided by that reasonable circumspection which the law expects from every one, and the want of which would precipitate a misfortune from a post or a wheelbarrow equally with a bicycle. The names attached to the bicycle-petition were signed because of the deliberate conviction that the danger from frightened horses was greater than from bicycles on the sidewalks. Horses in the country do not all get accustomed to these Centaur-like appearances, as those in the city do to corresponding monsters, such as the elevated trains, because different horses come into the village daily from the surrounding region. These animals standing on their hind legs, or rushing and shying, were frequent occurrences, whereas an unskilled bicycle-rider on the sidewalk can be avoided, and a skilled one can always avoid the pedestrian.
It is now, we hope, made apparent that Stockbridge barbarism has been prematurely assumed on ex parte testimony. Until a law should be passed to the contrary, the wide sidewalks away from the crowded part of the town were rightfully free to baby-carriages, children's wagons, wheelbarrows, and bicycles, and in the present state of the population there seem to have been good reasons why there should be no such law. At least, enough has been said to show that "the community" does not "acknowledge the outrage," and that the reason why the gentleman in question was "not supported but was condemned for his action" was, that it was reasonably regarded as an un-wise and unnecessary attempt to curtail those rural privileges which citizens commorant in the country as well as the villagers themselves regard as a great charm of their summer.
Inaccurate "data" and the proposal of unnecessary legislation come strangely from the disciples of Herbert Spencer. Strange, too, would it be if they were nearer the "barbarism" of their master's code than the orderly lovers of freedom whom they denounce. "Infinite presumption is discernible in this attempt at regulating the doings of men by law. . . . The desire to command is essentially a barbarous desire" (Spencer's "Social Statics," pp. 321, 80).
The length of this communication may seem monstrous. But the writer has been long-suffering. As a lover of peace, he had allowed the printed circular distributed throughout the village, which contained the same one-sided version of facts on which your article is based, as well as the impertinent and libelous attack in the "Springfield Republican," to go unanswered. For the good fame of Stockbridge, at which the professor, while leaving it for some years, it is understood, as a residence, aims through the editor of "The Popular Science Monthly" this Parthian shot, I could hardly say less. But, unless some astonishing twisting or suppression of material facts compel a further statement, I shall not again ask permission to trespass on your interesting columns.
Vermiculus Obtritus.
New York, December 22, 1882.
Retrieved from "https://en.wikisource.org/w/index.php?title=Popular_Science_Monthly/Volume_22/February_1883/Correspondence&oldid=8850384"
Popular Science Monthly Volume 22
Popular Science Monthly correspondence

redpajama_set_name: RedPajamaCommonCrawl


Abstract: The apparatus for providing a measured amount of a low boiling liquefied gas which can flow from an outlet of a thermally insulated container includes a fleece made of natural material, plastic material or metal connected with the outlet of the container, advantageously located immediately downstream of the outlet. This fleece provides a satisfactory separation of liquid and gas phases and thus a laminar liquefied gas flow and can assist in preventing freezing at the outlet due to moisture in the air. Advantageously a precise regulating valve which is sunk in the liquefied gas in the supply container provides continuous regulation of the liquefied gas flow.
a precision regulating valve submerged in said liquefied gas mounted adjacent the bottom of said container, said precision regulating valve being connected with said outlet and being connectable to a regulating device located outside of saidcontainer.
2. An apparatus according to claim 1 wherein said regulating device outside said container comprises an adjustable motor and said gap is from about 2 mm to about 30 mm.
3. In an apparatus for delivering a measured amount of a low-boiling liquefied gas which can flow from an outlet of a thermally insulated container, the improvement comprising a fleece made from a substance selected from the group consisting ofnatural material, plastic material and metal connected to said outlet of said container and wherein said fleece is surrounded by a substantially cylindrical shroud open at the end of said shroud furthest from said outlet so that a gap of from about 2 to30 mm is present between said fleece and said shroud.
4. The improvement according to claim 3 wherein a precision regulating valve submerged in said liquefied gas is mounted adjacent the bottom of said container, said precision regulating valve being connected with said outlet and with a regulatingdevice positioned outside of said container.
5. The improvement according to claim 4 wherein said regulating device comprises an adjustable motor.
(d) enclosing said cold gas in a shroud preventing access of ambient moisture to said fleece.
7. In an apparatus for delivering a measured amount of a low-boiling liquified gas which can flow from an outlet of a thermally insulated container, the improvement comprising a fleece made from a substance selected from the group consisting ofnatural material, plastic material and metal connected to said outlet of said container wherein said fleece is substantially cylindrically shaped and comes to a point at a downstream side of said fleece.
My present invention relates to an apparatus for delivering a measured amount of a low-boiling liquefied gas, especially liquid nitrogen.
An apparatus for delivering a measured amount of a low-boiling liquefied gas is known in which the liquefied gas can flow from an outlet of a thermally insulated cryogenic container.
Low-boiling liquefied gas is stored customarily in a thermally insulated cryogenic container and can be drawn off through an outlet. Particularly small quantities of low-boiling liquid gas may be measured out only with considerable difficultybecause the gas arising at the outlet interrupts the continuous liquefied gas flow. Hence expensive devices were developed to attain a continuous liquefied gas flow for a measured small quantity of liquefied gas.
Thus a device according to German Pat. No. 34 02 292 for delivering a measured small amount of low-boiling liquefied gas is known in which the liquefied gas is located in one container which is surrounded by a second container so that anintermediate space arises between the containers. The liquefied gas can be drawn from the container through an outlet. Cold gas issues from the inner to the outer container and cools the entire apparatus so that the liquefied gas found in the innercontainer receives a minimum of heat from the exterior. The cold gas leaves the container by a gas outlet in the vicinity of the fluid outlet for the liquefied gas so that it surrounds the liquefied gas flowing out of the container. To confine theliquefied gas flowing out, a gas cushion is formed under an elevated pressure corresponding at least to the total gas pressure existing in the surroundings in the vicinity of the outlet.
A device for delivery of an uninterrupted stream of a cryogenic liquefied gas is known and described in German Open Patent Application No. 34 19 855. This device is connected with a delivery pipe and a degassing device downstream by a pressureregulator containing the cryogenic liquefied gas. Free liquefied gas leaves the degassing device while cold gas is conducted to a different location. Consequently an unbroken stream of cryogenic liquefied gas can be generated.
All known devices for delivering a measured amount of a liquefied gas, particularly for a small quantity, which deliver a continuous fluid flow are comparatively expensive.
It is an object of my invention to provide an improved apparatus for delivering a measured amount of a liquefied gas which will obviate the aforementioned drawbacks.
It is also an object of my invention to provide an improved apparatus for delivering a measured amount of a liquefied gas, particularly a small amount, which can flow from an outlet of a thermally insulated cryogenic container constructed so thatwithout substantially increased expense an accurately measured laminar flow of liquefied gas can be drawn from the apparatus.
These objects and others which will become more readily apparent hereinafter are attained in accordance with my invention in an apparatus for delivering a measured amount of a liquefied gas which can flow from an outlet of a thermally insulatedcontainer.
According to my invention a fleece made of natural material, plastic or other synthetic material or metal is connected with and advantageously lies immediately downstream of the outlet of the container. A "fleece" in the sense of the invention,is a coherent body of matted fibers, filaments or threads or wires with interconnected interstices.
My invention includes a thermally insulated container, advantageously a cryogenic container such as a Dewar flask, for the liquefied gas. At the lower end of the container an outlet for the liquefied gas is provided with a fleece of naturalmaterial, plastic material or metal. The container is filled with liquefied gas, particularly liquid nitrogen. The liquefied gas is fed through the outlet to the plastic, metal or natural fleece and leaves the device in a laminar stream.
The liquefied gas found in the container flows through the outlet opening into the fleece. In the fleece a separation of the liquid and gas phase occurs so that a laminar flow of liquefied gas substantially vertically downward is delivered whilethe cold gas is conducted away laterally from the plastic, metal or natural fleece. The cold gas does not disturb the laminar liquefied gas flow so that a precise small amount of liquefied gas can be provided.
A variety of specific forms of my invention are possible. The fleece is advantageously substantially cylindrically shaped with a point on the end opposite the end adjacent or closest to the outlet of the container.
In this way a laminar downwardly directed liquefied gas stream is provided which is particularly suitable for delivering small amounts of liquefied gas. The cold gas is directed horizontally by the cylindrical fleece uniformly distributed overthe cylindrical surface.
In a particularly advantageous feature of my invention, the fleece is surrounded by a substantially cylindrical shroud open at its end furthest from the outlet so that a gap around the fleece of about 2 to about 30 mm (radial gap width) ispresent. This shroud ensures that the cold gas conducted away from the fleece will be deflected downwardly. So the shroud surrounds the plastic or metal fleece and the liquefied gas flowing downwardly. This prevents the admission of moist air to thefleece and thus freezing up and deposit of ice are prevented.
A precision regulating valve submerged in the liquefied gas is appropriately mounted adjacent the bottom of the container. This precision regulating valve is connected with the outlet and with a regulating device (i.e. a valve-controllingservomotor) located outside the container. This allows continuous metering of the amount of the liquefied gas issuing from the outlet and thus a precise control of the laminar liquefied fluid flow delivered from the fleece. Since the precise regulatingvalve is submerged in the liquefied gas freezing of the valve is prevented. The fine regulating valve can alternatively or in addition be controlled by hand by a simple rod or bar which is connected with the valve and projects from the container.
However it has proved especially suitable to control the precision regulating valve by the servomotor which is connected to it (i.e. the regulating device outside the container comprises a remotely controlled motor). In this way the apparatuscan be automated. Thus a predetermined amount of liquefied gas can be metered into a unit for food packaging, beverage filling or the like units located under the apparatus.
The apparatus according to my invention has in contrast to the conventional metering device for liquefied gases the advantage that a precise metering of small amounts of low-boiling liquefied gas is possible. A satisfactory separation of fluidand gas phases occurs in the fleece of my invention thus delivering a suitable laminar liquefied gas flow. A continuous regulation of the liquefied gas volume in the desired region is possible using the precise regulating valve submerged in theliquefied gas.
The apparatus according to my invention is suitable for delivering a measured small quantity of liquefied gas, particularly liquid nitrogen. Liquid nitrogen is used particularly in the food packaging industry. Nitrogen often serves as an inertgas in food packaging. Frequently the liquefied gas can produce a certain definite interior pressure after closing the package to increase the resistance or strength of the package, e.g. for stacking the packages over each other.
Accordingly with the apparatus of my invention a precise small quantity of liquefied gas, e.g. liquid nitrogen, can be provided to a food packaging unit. By operating a precise regulating valve with an adjustable motor, a predetermined amount ofliquefied nitrogen can automatically be provided, e.g. in a beverage filling machine with a high working speed.
The above and other objects, features and advantages of my invention will become more readily apparent from the following description, reference being made to the accompanying highly diagrammatic drawing in which the sole FIGURE is a schematicvertical cross sectional view of an apparatus for delivering a measured amount of a low-boiling liquefied gas according to my invention.
The apparatus in the drawing comprises a thermally insulated vacuum container or Dewar flask 1 which receives a low-boiling liquefied gas 2. The container or Dewar flask 1 is open on top so that fresh liquefied gas can be fed in from above andvapors can escape. By standard automatic regulation the liquefied gas level 5 can be maintained constant by control of the liquefied gas input.
A precision regulating valve 6 is positioned at the bottom lb of the Dewar or vacuum container 1 submerged in the liquefied gas 2. It is accessible to a regulating device R positioned outside of the container I shown in the drawing with a blockonly by a rigid connecting member 7. The precision regulating valve 6 can be controlled also by hand by the rigid connecting member 7.
An outlet 8 for the liquefied gas 2 is provided at the lower end of the Dewar or container 1. A pipe 9 for the liquefied gas 2 which is connected with the precision regulating valve 6 passes through the outlet 8 and ends at a fleece 10 made ofnatural material, plastic material or metal. The fleece 10 is cylindrical and comes to a point at its bottom so that a thin laminar liquefied gas stream 11 is delivered when the precision regulating valve 6 is open. The cold gas however escapeslaterally from the fleece 10.
The fleece 10 is surrounded by a shroud 13 so that a gap G around the fleece of about 15 mm is present between the fleece 10 and the shroud 13. Because of that cold gas 12 escaping laterally from the fleece 10 is deflected downwardly accordingto the arrows 14 and surrounds the fleece 10 so that admission of moist air is avoided and thus freezing is prevented.
In this example of my invention the regulating device R outside the vacuum container 1 comprises a servomotor with a remote controller (not shown) and the rigid connecting member 7 can be a connecting rod driven by the shaft of the adjustablemotor directly or through a transmission or gear box.
The width of the gap G between the shroud 13 and the fleece 10 can be from about 2 to about 30 mm.


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