Counting self-avoiding walks on free products of graphs

The connective constantμ(G) of a graph G is the asymptotic growth rate of the number σn of self-avoiding walks of length n in G from a given vertex. We prove a formula for the connective constant for free products of quasi-transitive graphs and show that σn∼AGμ(G)n for some constant AG that depends on G. In the case of products of finite graphs μ(G) can be calculated explicitly and is shown to be an algebraic number.