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arXiv · 2109.00409

On the $A_α$ spectral radius and $A_α$ energy of non-strongly connected digraphs

Xiuwen Yang·Ligong Wang
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Abstract

Let $A_α(G)$ be the $A_α$-matrix of a digraph $G$ and $λ_{α1}, λ_{α2}, \ldots, λ_{αn}$ be the eigenvalues of $A_α(G)$. Let $ρ_α(G)$ be the $A_α$ spectral radius of $G$ and $E_α(G)=\sum_{i=1}^n λ_{αi}^2$ be the $A_α$ energy of $G$ by using second spectral moment. Let $\mathcal{G}_n^m$ be the set of non-strongly connected digraphs with order $n$, which contain a unique strong component with order $m$ and some directed trees which are hung on each vertex of the strong component. In this paper, we characterize the digraph which has the maximal $A_α$ spectral radius and the maximal (minimal) $A_α$ energy in $\mathcal{G}_n^m$.

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Xiuwen Yang, Ligong Wang. 2021-09-01. On the $A_α$ spectral radius and $A_α$ energy of non-strongly connected digraphs. https://arxiv.org/abs/2109.00409

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