# Random walks and electrical resistances in products of graphs

## Abstract

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.

## Publication Title

Discrete Applied Mathematics

## Recommended Citation

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