On the $A_\alpha$ spectral radius of generalized weighted digraphs

Abstract

Let $G=(V(G), E(G))$ be a generalized weighted digraph without loops and multiple arcs, where the weight of each arc is a nonnegative and symmetric matrix of same order $p$.
For $v_i\in V(G)$, let $w_i^+=\sum\limits_{v_j\in N_i^{+}}w_{ij}$, where $w_{ij}$ is the weight of the arc $(v_i,v_j)$, and $N_i^{+}$ is the out-neighbors of the vertex $v_i$. Let $A_\alpha(G)=\alpha D(G)+(1-\alpha)A(G)$, where $0\leq \alpha\leq1$, $A(G)$ is the adjacency matrix of the generalized weighted digraph $G$, and $D(G)=diag(w_1^+,w_2^+,\ldots,w_n^+)$. The spectral radius of $A_\alpha(G)$ is called the $A_\alpha$ spectral radius of $G$.
In this paper, we give some upper bounds on the $A_\alpha$ spectral radius of generalized weighted digraphs. As application, we obtain some upper bounds on the $A_\alpha$ spectral radius of weighted digraphs and unweighted digraphs.