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

Abstract

Let $\mathcal{U}(n,k)$ and  $\Gamma(n,k)$ be respectively the sets of the $k$-uniform linear and nonlinear unicyclic hypergraphs having perfect matchings with  $n$ vertices, 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 $k\geq 3$.