Distance spectral radius of the complements of trees with fixed parameters

Abstract

Let G be a simple connected graph and Gc be the complement of G. Denote by dG(u,v) the distance between the vertices u and v which is the length of the shortest path between u and v in G. Let D(G)=(dG(u,v))n×n be the distance matrix of G and λ1(D(G)) denote the distance spectral radius of G. In this paper, we characterize the unique graphs with maximum distance spectral radius among all the complements of trees of order n with fixed maximum degree, pendent vertices, diameter and perfect matchings, respectively.