On the $D_\alpha$ spectral radius of strongly connected digraphs

Abstract

Let $G$ be a strongly connected digraph with distance matrix $D(G)$ and let $Tr(G)$ be the diagonal matrix with vertex transmissions of $G$. For any real $\alpha\in[0,1]$, define the matrix $D_\alpha(G)$ as
$$D_\alpha(G)=\alpha Tr(G)+(1-\alpha)D(G).$$
The $D_\alpha$ spectral radius of $G$ is the spectral radius
of $D_\alpha(G)$. In this paper, we first give some upper and lower bounds for the $D_\alpha$ spectral radius of $G$ and characterize the extremal digraphs. Moreover, for digraphs that are not transmission regular, we give a lower bound on the difference between the maximum vertex transmission and the $D_\alpha$ spectral radius. Finally, we obtain the $D_\alpha$ eigenvalues of the join of certain regular digraphs.