The distance spectrum and energy of the compositions of regular graphs

The distance energy of a graph GG is a recently developed energy-type invariant, defined as the absolute deviation of the eigenvalues of the distance matrix of GG. It is a useful molecular descriptor in QSPR modelling, as demonstrated by Consonni and Todeschini in [V. Consonni, R. Todeschini, New spectral indices for molecule description, MATCH Commun. Math. Comput. Chem. 60 (2008) 3–14]. We describe here the distance spectrum and energy of the join-based compositions of regular graphs in terms of their adjacency spectrum. These results are used to show that there exist a number of families of sets of noncospectral graphs with equal distance energy, such that for any n∈N, each family contains a set with at least nn graphs. The simplest such family consists of sets of complete bipartite graphs.