The generalized distance matrix of digraphs

Abstract

Let D(G) and DQ(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,…,Dn) be the diagonal matrix with vertex transmissions of the digraph G. To track the gradual change of D(G) into DQ(G), in this paper, we propose to study the convex combinations of D(G) and Diag(Tr) defined byDα(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 Dα(G) is called the Dα spectral radius of G, denoted by μα(G). We determine the digraph which attains the maximum (or minimum) Dα spectral radius among all strongly connected digraphs. Moreover, we also determine the digraphs which attain the minimum Dα spectral radius among all strongly connected digraphs with given parameters such as dichromatic number, vertex connectivity or arc connectivity.