RANDOM-WALKS AND ELECTRICAL RESISTANCES IN PRODUCTS OF GRAPHS

We study random walks and electrical resistances between pairs of vert ices in products of graphs. Among the results we prove are the followi ng. (1) In a graph G x P where P is a path with endvertices x and y, a nd G is any graph, with vertices a and b, the resistance behveen verti ces (a, x) and (b, v) is maximised at v = y. (2) In a graph G x K-n, f or vertices x and y of the complete graph K-n, 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.