Some Spectral Properties and Characterizations of Connected Odd-bipartite Uniform Hypergraphs

TL;DR

This paper explores spectral properties of connected odd-bipartite uniform hypergraphs, establishing conditions for spectral spectrum equality, characterizations, and properties of hypergraph products, advancing understanding of their spectral structure.

Contribution

It provides new spectral characterizations of connected odd-bipartite hypergraphs and answers open questions about their spectral radii and product properties.

Findings

Laplacian and signless Laplacian spectra are equal iff the hypergraph is odd-bipartite with even k

Characterization of hypergraphs with equal Laplacian and signless Laplacian spectral radii

Laplacian spectral radius of Cartesian product equals sum of individual spectral radii

Abstract

A $k$-uniform hypergraph $G=(V,E)$ is called odd-bipartite ([5]), if $k$ is even and there exists some proper subset $V_1$ of $V$ such that each edge of $G$ contains odd number of vertices in $V_1$. Odd-bipartite hypergraphs are generalizations of the ordinary bipartite graphs. We study the spectral properties of the connected odd-bipartite hypergraphs. We prove that the Laplacian H-spectrum and signless Laplacian H-spectrum of a connected $k$-uniform hypergraph $G$ are equal if and only if $k$ is even and $G$ is odd-bipartite. We further give several spectral characterizations of the connected odd-bipartite hypergraphs. We also give a characterization for a connected $k$-uniform hypergraph whose Laplacian spectral radius and signless Laplacian spectral radius are equal, thus provide an answer to a question raised in [9]. By showing that the Cartesian product $G\Box H$ of two odd-bipartite $k$-uniform hypergraphs is still odd-bipartite, we determine that the Laplacian spectral radius of $G\Box H$ is the sum of the Laplacian spectral radii of $G$ and $H$, when $G$ and $H$ are both connected odd-bipartite.