The ρ‐-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 M G ρ ( u ) = ∑ v ∈ V ρ ( v ) d i s t ( u , v ) , where d i s t ( . , . ) 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 Dalf o ´ 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).