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arXiv · 2109.03507

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

Peng-Li Zhang·Xiao-Dong Zhang
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Abstract

For $0\leq \alpha < 1$, 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.

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Peng-Li Zhang, Xiao-Dong Zhang. 2021-09-08. Lower bounds for the $\mathcal{A}_{\alpha}$-spectral radius of uniform hypergraphs. https://arxiv.org/abs/2109.03507

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