Restricted random walks on a graph

TL;DR

This paper develops a transfer matrix approach to analyze restricted random walks on graphs, deriving explicit generating functions that account for reversal steps, with applications to various lattice structures and the complete graph.

Contribution

It introduces a closed-form transfer matrix method for counting restricted random walks with reversals on any graph, linking results to eigenvalues of the adjacency matrix.

Findings

Explicit generating functions for walks with reversals on graphs.

Application of the method to periodic lattices and complete graphs.

Connection between generating functions and adjacency matrix eigenvalues.

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.