The connective constant mu(G) of an infinite transitive graph G is the exponential growth rate of the number of self-avoiding walks from a given origin. The relationship between connective constants and amenability is explored in the current work. Various properties of connective constants depend on the existence of so-called `unimodular graph height functions', namely: (i) whether mu(G) is a local function on certain graphs derived from G, (ii) the equality of mu(G) and the asymptotic growth rate of bridges, and (iii) whether there exists a terminating algorithm for approximating mu(G) to a given degree of accuracy. In the context of amenable groups, it is proved that the Cayley graphs of infinite, finitely generated, elementary amenable (and, more generally, virtually indicable) groups support unimodular graph height functions, which are in addition harmonic. In contrast, the Cayley graph of the Grigorchuk group, which is amenable but not elementary amenable, does not have a graph height function. In the context of non-amenable, transitive graphs, a lower bound is presented for the connective constant in terms of the spectral bottom of the graph. This is a strengthening of an earlier result of the same authors. Secondly, using a percolation inequality of Benjamini, Nachmias, and Peres, it is explained that the connective constant of a non-amenable, transitive graph with large girth is close to that of a regular tree. Examples are given of non-amenable groups without graph height functions, of which one is the Higman group. The emphasis of the work is upon the structure of Cayley graphs, rather than upon the algebraic properties of the underlying groups. New methods are needed since a Cayley graph generally possesses automorphisms beyond those arising through the action of the group.
