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On a Conjecture of Bollobás and Brightwell Concerning Random Walks on Product Graphs

Published online by Cambridge University Press:  01 December 1998

Department of Mathematics, Chalmers University of Technology, S-412 96 Göteborg, Sweden (e-mail:


We consider continuous time random walks on a product graph G×H, where G is arbitrary and H consists of two vertices x and y linked by an edge. For any t>0 and any a, bV(G), we show that the random walk starting at (a, x) is more likely to have hit (b, x) than (b, y) by time t. This contrasts with the discrete time case and proves a conjecture of Bollobás and Brightwell. We also generalize the result to cases where H is either a complete graph on n vertices or a cycle on n vertices.

Research Article
1998 Cambridge University Press

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