Counting Self-avoiding Walks

The connective constant μ(G) of a graph G is the asymptotic growth rate of the number of self-avoiding walks on G from a given starting vertex. We survey three aspects of the dependence of the connective constant on the underlying graph G. Firstly, when G is cubic, we study the effect on μ(G) of the Fisher transformation (that is, the replacement of vertices by triangles). Secondly, we discuss upper and lower bounds for μ(G) when G is regular. Thirdly, we present strict inequalities for the connective constants μ(G) of vertex-transitive graphs G, as G varies. As a consequence of the last, the connective constant of a Cayley graph of a finitely generated group decreases strictly when a new relator is added, and increases strictly when a non-trivial group element is declared to be a generator. Special prominence is given to open problems.