Moments in graphs

Let GG be a connected graph with vertex set VV and a weight function  ρρ that assigns a nonnegative number to each of its vertices. Then, the ρρ-moment   of GG at vertex uu is defined to be MGρ(u)=∑v∈Vρ(v)dist(u,v), where dist(⋅,⋅) stands for the distance function. Adding up all these numbers, we obtain the ρρ-moment of  GG: MGρ=∑u∈VMGρ(u)=12∑u,v∈Vdist(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)W(G), when ρ(u)=1/2ρ(u)=1/2 for every u∈Vu∈V, and the degree distance  D′(G)D′(G), obtained when ρ(u)=δ(u)ρ(u)=δ(u), the degree of vertex uu.In this paper we derive some exact formulas for computing the ρρ-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 ρρ-moments of its factors. As a consequence, we provide a method for obtaining nonisomorphic graphs with the same ρρ-moment for every ρρ (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.