The $H$-spectrum of a generalized power hypergraph

TL;DR

This paper characterizes the $H$-spectra of generalized power hypergraphs, linking their eigenvalues to those of subgraphs of the original graph, and explores the spectral limits of such hypergraphs.

Contribution

It provides a complete description of the $H$-spectra of certain hypergraphs in terms of subgraph eigenvalues, extending spectral graph theory to hypergraphs.

Findings

The $H$-spectrum of $G^{k,k/2}$ is determined by eigenvalues of subgraphs of $G$.

$G^{k,k/2}$ shares the same least eigenvalues as $G$.

Constructs hypergraphs with least eigenvalues converging to $-\sqrt{2+\sqrt{5}}$.

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 $G^{k,{k \over 2}}$ is non-odd-bipartite if and only if $G$ is non-bipartite, and $G^{k,{k \over 2}}$ 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, $G^{k,{k \over 2}}$ 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 $-\sqrt{2+\sqrt{5}}$.