The generalized distance matrix of digraphs

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)=diag(D₁,D₂,…,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≤α≤1. 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.