Some results on extremal spectral radius of hypergraph

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 ($k\ge 3$) 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 ($k\ge 3$) unicyclic hypergraphs with fixed number of vertices and fixed girth, the hypergraphs with the maximum spectral radius are completely determined; among all $k$-uniform ($k\ge 3$) $octopuslike$ hypergraphs with fixed number of vertices, the hypergraphs with the minimum spectral radius are completely determined. As well, for $k$-uniform ($k\ge 3$) $lollipop$ hypergraphs, we get that the spectral radius decreases with the girth increasing.

arXiv admin note: substantial text overlap with arXiv:2306.10184, arXiv:2306.16027