Resistance distances and Kirchhoff index of graphs with an involution

Motivated by the work of Zhang and Yan (2009), this paper considers the problem of computing resistance distances and Kirchhoff index of graphs with an involution. We show that if G is a weighted graph with an involution, then the resistance distance and the Kirchhoff index of G can be expressed in terms of parameters of two weighted graphs with a smaller size. As applications, we compute resistance distances and Kirchhoff indices of double graphs, the almost-complete graph and the almost-complete bipartite graph.