Random walks and electrical resistances in products of graphs
We study random walks and electrical resistances between pairs of vertices in products of graphs. Among the results we prove are the following. (1) In a graph G × P, where P is a path with endvertices x and y, and G is any graph, with vertices a and b, the resistance between vertices (a, x) and (b, v) is maximised at v = y. (2) In a graph G × Kn, for vertices x and y of the complete graph Kn and a, b of the graph G, the probability that a random walk, starting from (a, x), reaches (b, x) before (b, y) is at least 1/2.
Discrete Applied Mathematics
Bollobás, B., & Brightwell, G. (1997). Random walks and electrical resistances in products of graphs. Discrete Applied Mathematics, 73 (1), 69-79. https://doi.org/10.1016/S0166-218X(96)00002-9