The maximal spectral radius of the uniform unicyclic hypergraph with perfect matchings

TL;DR

This paper determines the hypergraphs with the largest spectral radius among certain classes of uniform unicyclic hypergraphs with perfect matchings, using transformation techniques and incidence matrix constructions.

Contribution

It identifies the maximal spectral radius hypergraphs within three specific classes of uniform unicyclic hypergraphs with perfect matchings, extending previous spectral graph theory results.

Findings

Maximal spectral radius hypergraphs for -uniform linear unicyclic hypergraphs with specific vertex counts.

Maximal spectral radius hypergraphs for -uniform nonlinear unicyclic hypergraphs with perfect matchings.

Unified characterization of maximal spectral radius hypergraphs across different classes.

Abstract

Let $\mathcal{U}(n,k)$ and $\Gamma(n,k)$ be the set of the $k$-uniform linear and nonlinear unicyclic hypergraphs having perfect matchings with $n$ vertices respectively, where $n\geq k(k-1)$ and $k\geq 3$. By using some techniques of transformations and constructing the incidence matrices for the hypergraphs considered, we get the hypergraphs with the maximal spectral radii among three kinds of hypergraphs, namely $\mathcal{U}(n,k)$ with $n= 2k(k-1)$ and $n\geq 9k(k-1)$, $\Gamma(n,k)$ with $n\geq k(k-1)$, and $\mathcal{U}(n,k)\cup \Gamma(n,k)$ with $n\geq 2k(k-1)$, where $k\geq 3$.