The proof of a conjecture on largest Laplacian and signless Laplacian H-eigenvalues of uniform hypergraphs
Let $\mathcal{A(}G\mathcal{)},\mathcal{L(}G\mathcal{)}$ and $\mathcal{Q(}% G\mathcal{)}$ be the adjacency tensor, Laplacian tensor and signless Laplacian tensor of uniform hypergraph $G$, respectively. Denote by $\lambda (\mathcal{T})$ the largest H-eigenvalue of tensor $\mathcal{T}$. Let $H$ be a uniform hypergraph, and $H^{\prime}$ be obtained from $H$ by inserting a new vertex with degree one in each edge. We prove that $\lambda(\mathcal{Q(}% H^{\prime}\mathcal{)})\leq\lambda(\mathcal{Q(}H\mathcal{)}).$ Denote by $G^{k}$ the $k$th power hypergraph of an ordinary graph $G$ with maximum degree $\Delta\geq2$. We will prove that $\{\lambda(\mathcal{Q(}% G^{k}\mathcal{)})\}$ is a strictly decreasing sequence, which imply Conjectrue 4.1 of Hu, Qi and Shao in \cite{HuQiShao2013}. We also prove that $\lambda(\mathcal{Q(}G^{k}\mathcal{)})$ converges to $\Delta$ when $k$ goes to infinity. The definiton of $k$th power hypergraph $G^{k}$ has been generalized as $G^{k,s}.$ We also prove some eigenvalues properties about $\mathcal{A(}% G^{k,s}\mathcal{)},$ which generalize some known results. Some related results about $\mathcal{L(}G\mathcal{)}$ are also mentioned.