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

The Kirchhoff index K(G) of a graph G is one of the most important topological indices in mathematical chemistry. Suppose that G is a unicyclic graph with n vertices and the cycle in G is denoted by C-k, where k is the length of this cycle. Let S(G) be the unicyclic graph obtained from G by inserting a new vertex to each edge of G. In this paper, we prove that the Kirchhoff index of G equals [m(S(G), n - 2) - 2m(S(G) - C-2k, n - k - 2)] /k if 3 <= k <= n - 2 and equals m(S(G), n - 2)/k if k = n - 1, n, where m(S(G), n - 2) is the number of matchings of S(G) with n - 2 edges and S(G) - C-2k is the acyclic graph obtained from S(G) by deleting all vertices of the cycle C-2k in S(G). (C) 2021 Elsevier B.V. All rights reserved.