The [formula omitted]‐-moments of vertex‐weighted graphs

Abstract

Let (G,ρ) be a vertex-weighted graph of G together with the vertex set V and a function ρ(V). A ρ-moment of G at a given vertex u is defined as MGρ(u)=∑v∈Vρ(v)dist(u,v), where dist(.,.) stands for the distance function. The ρ-moment of G is the sum of moments of all vertices in G. This parameter is closely related to degree distance, Wiener index, Schultz index etc. Motivated by earlier work of Dalfo´ et al. (2013), we introduce three classes of hereditary graphs by vertex(edge)-grafting operations and give the expressions for computing their ρ-moments, by which we compute the ρ-moments of uniform(non-uniform) cactus chains and derive the order relations of ρ-moments of uniform(non-uniform) cactus chains. Based on these relations, we discuss the extremal value problems of ρ-moments in biphenyl and polycyclic hydrocarbons, and extremal polyphenyl chains, extremal spiro chains etc are given, respectively. This generalizes the results of Deng (2012).