On some degree-and-distance-based graph invariants of trees

Let G be a connected graph with vertex set V(G). For u, v ∈ V(G), d(v) and d(u, v) denote the degree of the vertex v and the distance between the vertices u and v. A much studied degree-and-distance-based graph invariant is the degree distance, defined as DD=∑_{u,v}⊆V(G)[d(u)+d(v)]d(u,v). A related such invariant (usually called "Gutman index") is ZZ=∑_{u,v}⊆V(G)[d(u)·d(v)]d(u,v). If G is a tree, then both DD and ZZ are linearly related with the Wiener index W=∑_{u,v}⊆V(G)d(u,v). We examine the difference DD-ZZ for trees and establish a number of regularities.