BOUNDS FOR THE Aα-SPECTRAL RADIUS OF A DIGRAPH

Abstract

Let $D$ be a digraph of order $n$ and let $A(D)$ be the adjacency matrix of $D$. Let $Deg(D)$ be thediagonal matrix of vertex out-degrees of $D$. For any real $\alpha \in [0,1],$ the generalized adjacency matrix $A_{\alpha }(D)$ of the $D$ is defined as $A_{\alpha }(D)=\alpha Deg(D)+(1-\alpha )A(D).$The largest modulus of the eigenvalues of $A_{\alpha }(D)$ is called the generalized adjacency spectral radius or the $A_{\alpha }$-spectralradius of $D$. In this paper, we obtain some new upper and lower bounds for the spectralradius of $A_{\alpha }(D)$ in terms of the number of vertices $n$, the number of arcs, the vertex out-degrees, the average 2-out-degrees of the vertices of $D$ and the parameter~$\alpha$. We characterize the extremal digraphs attaining these bounds.