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

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 $k\geq 3$.