Annals of Combinatorics 3 (1999) 475-481
Restricted Random Walks on a Graph
F.Y. Wu and H. Kunz
Department of Physics, Northeastern University, Boston, MA 02115, USA
Institut de Physique Théorique, Ecole Polytechnique Fédérale, Lausanne, Switzerland
Received November 1, 1998
AMS Subject Classification: 60J15, 82B41
Abstract. The problem of a restricted random walk on graphs, which keeps track of the number of immediate reversal steps, is considered by using a transfer matrix formulation. A closed-form expression is obtained for the generating function of the number of n-step walks with r reversal steps for walks on any graph. In the case of graphs of a uniform valence, we show that our result has a probabilistic meaning, and deduce explicit expressions for the generating function in terms of the eigenvalues of the adjacency matrix. Applications to periodic lattices and the complete graph are given.
Keywords: random walks, graphs, adjacency matrix
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