The distance spectrum and energy of the compositions of regular graphs

Distance energy of a graph G is a recent energy-type invariants, defined as the absolute deviation of the eigenvalues of the distance matrix of G. It is a useful molecular descriptor in QSPR modelling, as demonstrated by Consonni and Todeschini in [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 the terms of their adjacency spectrum. These results are used to show that there exists 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 n graphs. The simplest such family consists of sets of complete bipartite graphs. MSC Classification: 05C50