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

Let G be a connected graph with vertex set V(G). For u, v ź 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 D D = ź { u , v } ź V ( G ) d ( u ) + d ( v ) d ( u , v ) . A related such invariant (usually called "Gutman index") is Z Z = ź { u , v } ź V ( G ) d ( u ) ź d ( v ) d ( u , v ) . If G is a tree, then both DD and ZZ are linearly related with the Wiener index W = ź { u , v } ź V ( G ) d ( u , v ) . We examine the difference D D - Z Z for trees and establish a number of regularities.