Lower bounds for the ${\mathcal{A}}_{\alpha }$-spectral radius of uniform hypergraphs

Abstract

For $0\le \alpha <1$, the ${\mathcal{A}}_{\alpha }$-spectral radius of a $k$-uniform hypergraph $G$ is defined to be the spectral radius of the tensor ${\mathcal{A}}_{\alpha }(G):=\alpha \mathcal{D}(G)+(1-\alpha )\mathcal{A}(G)$, where $\mathcal{D}(G)$ and $A(G)$ are diagonal and the adjacency tensors of $G$ respectively. This paper presents several lower bounds for the difference between the ${\mathcal{A}}_{\alpha }$-spectral radius and an average degree $\frac{km}{n}$ for a connected $k$-uniform hypergraph with $n$ vertices and $m$ edges, which may be considered as the measures of irregularity of $G$. Moreover, two lower bounds on the ${\mathcal{A}}_{\alpha }$-spectral radius are obtained in terms of the maximum and minimum degrees of a hypergraph.

22 pages

References in corpus (1)

• Perron-Frobenius theorem for nonnegative multilinear forms and extensions