On the distance spectral radius of trees

Abstract

In this article, we prove that among trees on n vertices and matching number m, the dumbbell is the unique tree that maximizes the distance spectral radius, which was conjectured by Aleksandar Ilić [Distance spectral radius of trees with given matching number, Discr. Appl. Math. 158 (2010), pp. 1799–1806]. In addition, we find the unique tree that maximizes the distance spectral radius in the class of trees with a given number of pendent vertices.