The generalized distance matrix of digraphs

Abstract

Let $D(G)$ and $D^Q(G)= Diag(Tr) + D(G)$ be the distance matrix and distance signless Laplacian matrix of a simple strongly connected digraph $G$, respectively, where $Diag(Tr)=\textrm{diag}(D_1,D_2,$ $\ldots,D_n)$ be the diagonal matrix with vertex transmissions of the digraph $G$. To track the gradual change of $D(G)$ into $D^Q(G)$, in this paper, we propose to study the convex combinations of $D(G)$ and $Diag(Tr)$ defined by $$D_α(G)=αDiag(Tr)+(1-α)D(G), \ \ 0\leq α\leq1.$$ This study reduces to merging the distance spectral and distance signless Laplacian spectral theories. The eigenvalue with the largest modulus of $D_α(G)$ is called the $D_α$ spectral radius of $G$, denoted by $μ_α(G)$. We determine the digraph which attains the maximum (or minimum) $D_α$ spectral radius among all strongly connected digraphs. Moreover, we also determine the digraphs which attain the minimum $D_α$ spectral radius among all strongly connected digraphs with given parameters such as dichromatic number, vertex connectivity or arc connectivity.