The largest $\alpha$-spectral radius of the $k$-uniform unicyclic hypergraphs with perfect matchings

Abstract

Let $G$ be a $k$-uniform hypergraph with $k\geqslant 2$ and $0\leqslant \alpha< 1$. The $\alpha$-spectral radius of $G$ is the largest modulus of all the eigenvalues of $\bm{\mathcal A}_\alpha(G)$, where $\bm{\mathcal A}_\alpha(G)=\alpha \bm{\mathcal D}(G) + (1 -\alpha)\bm{\mathcal A}(G)$ is the convex linear combination of $\bm{\mathcal D}(G)$ and $\bm{\mathcal A}(G)$ with $\bm{\mathcal D}(G)$ and $\bm{\mathcal A}(G)$ being the degree diagonal tensor and the adjacency tensor of $G$, respectively. Let $\mathcal{U}(n,k)$ be the set of the $k$-uniform unicyclic hypergraphs having perfect matchings with $n$ vertices, where $n\geqslant k(k-1)$ and $k\geqslant 3$.
By using a creative method of the $\alpha$-Perron vector and several techniques for studying the $\alpha$-spectral radii of hypergraphs, such as the well-known Perron--Frobenius theorem, the moving-edge operation, and the 2-switch transformation, the hypergraph with the largest $\alpha$-spectral radius is characterized among $\mathcal{U}(n,k)$, where $n\geqslant k(k-1)$ and $k\geqslant 3$.