Combinatorial explanation of the weighted Wiener (Kirchhoff) index of trees and unicyclic graphs

Abstract

Let G be a vertex-weighted graph with vertex-weighted function ω:V(G)→R+ satisfying ω(vi)=xi for each vi∈V(G)={v1,v2,…,vn}. The weighted Wiener index of vertex-weighted graph G is defined as W(G;x1,x2,…,xn)=∑1≤i<j≤nxixjdG(vi,vj), where dG(vi,vj) denotes the distance between vi and vj in G. In this paper, we give a combinatorial explanation of W(G;x1,x2,…,xn) when G is a tree: W(G;x1,x2,…,xn)=m(S(G),n−2)∏i=1nxi, where m(S(G),n−2) is the sum of weights of matchings of some weighted subdivision S(G) of G with n−2 edges. We also give a similar combinatorial explanation of the weighted Kirchhoff index K(G;x1,x2,…,xn)=∑1≤i<j≤nxixjrG(vi,vj) of a unicyclic graph G, where rG(vi,vj) denotes the resistance distance between vi and vj in G. These results generalize some known formulae on the Wiener and Kirchhoff indices.