On distance integral graphs

Abstract

The distance eigenvalues of a connected graph G are the eigenvalues of its distance matrix D, and they form the distance spectrum of G. A graph is called distance integral if its distance spectrum consists entirely of integers. We show that no nontrivial tree can be distance integral. We characterize distance integral graphs in the classes of graphs similar to complete split graphs, which, together with relations between graph operations and distance spectra, allows us to exhibit many infinite families of distance integral graphs.