Calculating the degree distance of partial Hamming graphs

Abstract

If G is a connected graph with vertex set V(G), then the degree distance of G is defined as DD{G) = Σ{u,v}∈V(G)(deg u + deg v)d{u,v), where deg u is the degree of vertex u, and d(u,v) denotes the distance between u and v. In the chemical Uterature, DD(G) is better known under the name Schultz index, introduced by Gutman. In the class of partial Hamming graphs, which include trees, benzenoid and phenylene systems among others, we express the degree distance in terms of the canonical metric representation, thus simplifying its computation.