The proof of a conjecture on largest Laplacian and signless Laplacian H-eigenvalues of uniform hypergraphs

Abstract

Let $\mathcal{A}(G),\mathcal{L}(G)$ and $\mathcal{Q}($ 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 $\lambda (\mathcal{Q}($ Denote by $G^k$ the $k$th power hypergraph of an ordinary graph $G$ with maximum degree $\Delta \ge 2$. We will prove that $\{\lambda (\mathcal{Q}($ is a strictly decreasing sequence, which imply Conjectrue 4.1 of Hu, Qi and Shao in \cite{HuQiShao2013}. We also prove that $\lambda (\mathcal{Q}(G^k))$ 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 $\mathcal{A}($ which generalize some known results. Some related results about $\mathcal{L}(G)$ are also mentioned.

References in corpus (1)

• The extremal spectral radii of $k$-uniform supertrees