Random flights on regular graphs

Abstract

Let K be a finite graph with vertex set V = {x₀, x₁, …, x_σ–1} and automorphism group G. It is assumed that G acts transitively on V. We can imagine that the vertices of K represent σ cities and a traveler visits the cities in a series of random flights. The traveler starts at a given city and in each flight, independently of the past journey, chooses a city at random as the destination. Denote by v_n (n = 1, 2, …) the location of the traveler at the end of the nth flight, and by v₀ the initial location. It is assumed that the transition probabilities P{v_n = x_j | v_n–1 = x_i}, x_i ϵ V, x_j ϵ V, do not depend on n and are invariant under the action of G on V. The main result of this paper consists in determining p(n), the probability that the traveler returns to the initial position at the end of the nth flight.