Moments in graphs

Let G be a connected graph with vertex set V and a weight function ρ that assigns a nonnegative number to each of its vertices. Then, the ρ-moment of G at vertex u is defined to be M G(u) = ∑ v∈V ρ(v) dist(u, v), where dist(·, ·) stands for the distance function. Adding up all these numbers, we obtain the ρ-moment of G: M G = ∑ u∈V M G(u) = 1 2 ∑ u,v∈V dist(u, v)[ρ(u) + ρ(v)]. This parameter generalizes, or it is closely related to, some well-known graph invariants, such as the Wiener index W (G), when ρ(u) = 1/2 for every u ∈ V , and the degree distance D′(G), obtained when ρ(u) = δ(u), the degree of vertex u. In this paper we derive some exact formulas for computing the ρ-moment of a graph obtained by a general operation called graft product, which can be seen as a generalization of the hierarchical product, in terms of the corresponding ρ-moments of its factors. As a consequence, we provide a method for obtaining nonisomorphic graphs with the same ρ-moment for every ρ (and hence with equal mean distance, Wiener index, degree distance, etc). In the case when the factors are trees and/or cycles, techniques from linear algebra allow us to give formulas for the degree distance of their product.