Resistance distance, Kirchhoff index, and Kemeny's constant in flower graphs

Abstract: We obtain a general formula for the resistance distance (or effective resistance) between any pair of nodes in a general family of graphs which we call flower graphs. Flower graphs are obtained from identifying nodes of multiple copies of a given base graph in a cyclic way. We apply our general formula to two specific families of flower graphs, where the base graph is either a complete graph or a cycle. We also obtain bounds on the Kirchhoff index and Kemeny's constant of general flower graphs using our formula for resistance. For flower graphs whose base graph is a complete graph or a cycle, we obtain exact, closed form expressions for the Kirchhoff index and Kemeny's constant.