On the Kirchhoff index of a graph and the matchings of the subdivision

Abstract

Suppose that G is a connected simple graph with n vertices. Let Kf(G) and m(G,i) denote the Kirchhoff index of G and the number of i-matchings of G, respectively. First, in 2006, Yan and Yeh showed that if G is a tree, then Kf(G)=m(S(G),n−2),where S(G) is the subdivision of G. Then, in 2021, Chen and Yan showed that if G is a unicyclic graph with the cycle Ck, then Kf(G)=[m(S(G),n−2)−2m(S(G)−S(Ck),n−2−k)]/k.In this paper, we generalize the above results to any connected simple graph. In fact, we obtain the expressions for all Laplacian coefficients of G in terms of m(S(G),i), which provides much more information about relations between various parameters of G and the matchings of the subdivision S(G).