Some results on extremal spectral radius of hypergraph

TL;DR

This paper investigates the spectral radius of hypergraphs, deriving formulas linking it to ordinary graphs, and determines extremal hypergraphs with respect to spectral radius under various structural constraints.

Contribution

The paper provides explicit formulas connecting hypergraph spectral radius to graph spectra and characterizes extremal hypergraphs for fixed parameters.

Findings

Identified hypergraphs with minimum and second minimum spectral radius among unicyclic hypergraphs.

Determined hypergraphs with maximum spectral radius given fixed girth.

Showed spectral radius decreases as girth increases in lollipop hypergraphs.

Abstract

For a $hypergraph$ $\mathcal{G}=(V, E)$ with a nonempty vertex set $V=V(\mathcal{G})$ and an edge set $E=E(\mathcal{G})$, its $adjacency$ $matrix$ $\mathcal {A}_{\mathcal{G}}=[(\mathcal {A}_{\mathcal{G}})_{ij}]$ is defined as $(\mathcal {A}_{\mathcal{G}})_{ij}=\sum_{e\in E_{ij}}\frac{1}{|e| - 1}$, where $E_{ij} = \{e\in E\, |\, i, j \in e\}$. The $spectral$ $radius$ of a hypergraph $\mathcal{G}$, denoted by $\rho(\mathcal {G})$, is the maximum modulus among all eigenvalues of $\mathcal {A}_{\mathcal{G}}$. In this paper, we get a formula about the spectral radius which link the ordinary graph and the hypergraph, and represent some results on the spectral radius changing under some graphic structural perturbations. Among all $k$-uniform ($k\geq 3$) unicyclic hypergraphs with fixed number of vertices, the hypergraphs with the minimum, the second the minimum spectral radius are completely determined, respectively; among all $k$-uniform ($k\geq 3$) unicyclic hypergraphs with fixed number of vertices and fixed girth, the hypergraphs with the maximum spectral radius are completely determined; among all $k$-uniform ($k\geq 3$) $octopuslike$ hypergraphs with fixed number of vertices, the hypergraphs with the minimum spectral radius are completely determined. As well, for $k$-uniform ($k\geq 3$) $lollipop$ hypergraphs, we get that the spectral radius decreases with the girth increasing.