The -spectra of a class of generalized power hypergraphs

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• 作者：Murad-ul-Islam Khan ^{muradulislam@foxmail.com" class="auth_mail" title="E-mail the corresponding author} ; Yi-Zheng Fan ; ^{fanyz@ahu.edu.cn" class="auth_mail" title="E-mail the corresponding author} ; Ying-Ying Tan ^{tansusan1@ahjzu.edu.cn" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Hypergraph ; Adjacency tensor ; Signless Laplacian tensor ; Least eigenvalue ; Limit point
• 刊名：Discrete Mathematics
• 出版年：2016
• 出版时间：6 June 2016
• 年：2016
• 卷：339
• 期：6
• 页码：1682-1689
• 全文大小：430 K

文摘

The generalized power of a simple graph G$G$, denoted by G^k,s$G^{k,s}$, is obtained from G$G$ by blowing up each vertex into an 70a5a17714e9a28b478" title="Click to view the MathML source">s$s$-set and each edge into a k$k$-set, where $1\le s\le \frac{k}{2}$. When 70c61f1506">$s<\frac{k}{2}$, G^k,s$G^{k,s}$ is always odd-bipartite. It is known that 70fd0">$G^{k,\frac{k}{2}}$ is non-odd-bipartite if and only if G$G$ is non-bipartite, and 70fd0">$G^{k,\frac{k}{2}}$ has the same adjacency (respectively, signless Laplacian) spectral radius as G$G$. In this paper, we prove that, regardless of multiplicities, the H$H$-spectrum of $\mathcal{A}\left(G^{k,\frac{k}{2}}\right)$ (respectively, $\mathcal{Q}\left(G^{k,\frac{k}{2}}\right)$) consists of all eigenvalues of the adjacency matrices (respectively, the signless Laplacian matrices) of the connected induced subgraphs (respectively, modified induced subgraphs) of G$G$. As a corollary, 70fd0">$G^{k,\frac{k}{2}}$ has the same least adjacency (respectively, least signless Laplacian) H$H$-eigenvalue as G$G$. We also discuss the limit points of the least adjacency H$H$-eigenvalues of hypergraphs, and construct a sequence of non-odd-bipartite hypergraphs whose least adjacency H$H$-eigenvalues converge to 42199cbb34f38e228591373d5adebb">$-\sqrt{2+\sqrt{5}}$.