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

Let G be a vertex-weighted graph with vertex-weighted function omega : V (G) -> R+ satisfying omega(v(i)) = x(i) for each v(i) is an element of V(G) = {v(1), v(2), ..., v(n)}. The weighted Wiener index of vertex-weighted graph G is defined as W (G; x(1), x(2), ..., x(n)) = Sigma(1 <= i <= j <= n) x(i)x(j)dG (v(i), v(j)), where d(G)(v(i), v(j)) denotes the distance between v(i) and v(j) in G. In this paper, we give a combinatorial explanation of W (G; x(1), x(2), ..., x(n)) when G is a tree: W (G; x(1), x(2), ..., x(n)) = m(S(G),n - 2) Pi(n)(i=1)x(i), 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; x(1), x(2) , ..., x(n)) = Sigma(1 <= i <= j <= n )x(i)x(j)rG(()v(i), v(j)) of a unicyclic graph G, where r(G)(v(i), v(j)) denotes the resistance distance between v(i) and v(j) in G. These results generalize some known formulae on the Wiener and Kirchhoff indices. (C) 2022 Elsevier B.V. All rights reserved.