The unicyclic hypergraph with extremal spectral radius

TL;DR

This paper identifies the unicyclic hypergraphs with the highest and second-highest spectral radii among all fixed-vertex, k-uniform hypergraphs, advancing understanding of spectral extremal properties in hypergraph theory.

Contribution

It determines the hypergraphs with the maximum and second maximum spectral radius among all fixed-vertex, k-uniform unicyclic hypergraphs.

Findings

Identified the hypergraph with the maximum spectral radius.

Identified the hypergraph with the second maximum spectral radius.

Abstract

For a $hypergraph$ $\mathcal{G}=(V, E)$ consisting of 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, among all $k$-uniform ($k\geq 3$) unicyclic hypergraphs with fixed number of vertices, the hypergraphs with the maximum and the second the maximum spectral radius are completely determined, respectively.