Ordering digraphs with maximum outdegrees by their $A_α$ spectral radius
Zengzhao Xu, Weige Xi, Ligong Wang
TL;DR
This work studies ordering of strongly connected digraphs by the $A_\alpha$ spectral radius $\lambda_\alpha(G)$ for $\alpha\in[\tfrac{1}{2},1)$. It provides two key upper bounds on $\lambda_\alpha(G)$ in terms of the maximum outdegree and arc/vertex counts, leveraging standard lemmas that relate $\lambda_\alpha(G)$ to local degree data. The main results show that, under suitable conditions on $\Delta^+(G)$ and the arc count $m$, one digraph with larger maximum outdegree has strictly larger $\lambda_\alpha$ (Theorem 1.1) or exceeds the other by at least $\tfrac{1}{4}$ (Theorem 1.2); a corollary strengthens the comparison for larger $\alpha$. These findings extend ordering results from undirected graphs to digraphs via the $A_\alpha$ matrix, providing concrete criteria for comparing digraphs by their outdegree structure.
Abstract
Let $G$ be a strongly connected digraph with $n$ vertices and $m$ arcs. For any real $α\in[0,1]$, the $A_α$ matrix of a digraph $G$ is defined as $$A_α(G)=αD(G)+(1-α)A(G),$$ where $A(G)$ is the adjacency matrix of $G$ and $D(G)$ is the outdegrees diagonal matrix of $G$. The eigenvalue of $A_α(G)$ with the largest modulus is called the $A_α$ spectral radius of $G$, denoted by $λ_α(G)$. In this paper, we first obtain an upper bound on $λ_α(G)$ for $α\in[\frac{1}{2},1)$. Employing this upper bound, we prove that for two strongly connected digraphs $G_1$ and $G_2$ with $n\ge4$ vertices and $m$ arcs, and $α\in [\frac{1}{\sqrt{2}},1)$, if the maximum outdegree $Δ^+(G_1)\ge 2α(1-α)(m-n+1)+2α$ and $Δ^+(G_1)>Δ^+(G_2)$, then $λ_α(G_1)>λ_α(G_2)$. Moreover, We also give another upper bound on $λ_α(G)$ for $α\in[\frac{1}{2},1)$. Employing this upper bound, we prove that for two strongly connected digraphs with $m$ arcs, and $α\in[\frac{1}{2},1)$, if the maximum outdegree $Δ^+(G_1)>\frac{2m}{3}+1$ and $Δ^+(G_1)>Δ^+(G_2)$, then $λ_α(G_1)+\frac{1}{4}>λ_α(G_2)$.