Resistance distances and Kirchhoff indices in graph joins

Abstract We present a combinatorial method for computing effective resistances and Kirchhoff indices in joins of two graphs. The method is based on winning probabilities in a game that we call k - Pass the Buck . We illustrate our approach across various families of graphs, including path joins, cycle joins, complete split graphs, cone graphs, and others. Finally, we examine Kemeny’s constants for random walks on joins of regular graphs.