The $H$-spectrum of a generalized power hypergraph

Abstract

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

arXiv admin note: text overlap with arXiv:1408.3303

References in corpus (3)

• Maximizing spectral radii of uniform hypergraphs with few edges
• Connected Hypergraphs with Small Spectral Radius
• Multiplicities of eigenvalues of tensors

Cited by in corpus (9)

• The spectral symmetry of weakly irreducible nonnegative tensors and connected hypergraphs
• Eigenvectors of Laplacian or signless Laplacian of Hypergraphs Associated with Zero Eigenvalue
• The spectrum of a class of uniform hypergraphs
• The largest $H$-eigenvalue and spectral radius of Laplacian tensor of non-odd-bipartite generalized power hypergraphs
• The cyclic index of adjacency tensor of generalized power hypergraphs
• Computing The Analytic Connectivity of A Uniform Hypergraph
• The least H-eigenvalue of signless Laplacian of non-odd-bipartite hypergraphs
• The least H-eigenvalue of adjacency tensor of hypergraphs with cut vertices
• The proof of a conjecture on largest Laplacian and signless Laplacian H-eigenvalues of uniform hypergraphs