Large sets of long distance equienergetic graphs

Abstract

Distance energy of a graph is a recent energy-type invariant, defined as the absolute deviation of the eigenvalues of the distance matrix of the graph. Two graphs of the same order are said to be distance equienergetic if they have equal distance energy, while they have distinct spectra of their distance matrices. Examples of pairs of distance equienergetic graphs appear in the literature already, but most of them have diameter two only.
We describe here the distance spectrum of a special composition of regular graphs, and, as an application, we show that for any n ≥ 3, there exists a set of n + 1 distance equienergetic graphs which have order 6n and diameter n − 1 each.