The ρ-moments of vertex-weighted graphs

Let (G, rho) be a vertex-weighted graph of G together with the vertex set V and a function rho(V). A rho-moment of G at a given vertex u is defined as M-G(rho)(u) = Sigma(v is an element of V) rho(v)dist(u, v), where dist(., .) stands for the distance function. The rho-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 rho-moments, by which we compute the rho-moments of uniform(nonuniform) cactus chains and derive the order relations of rho-moments of uniform(nonuniform) cactus chains. Based on these relations, we discuss the extremal value problems of rho-moments in biphenyl and polycyclic hydrocarbons, and extremal polyphenyl chains, extremal spiro chains etc are given, respectively. This generalizes the results of Deng (2012). (C) 2021 Elsevier Inc. All rights reserved.