Strict Inequalities for Connective Constants of Transitive Graphs

The connective constant of a graph is the exponential growth rate of the number of self-avoiding walks starting at a given vertex. Strict inequalities are proved for connective constants of vertex-transitive graphs. First, the connective constant decreases strictly when the graph is replaced by a nontrivial quotient graph. Second, the connective constant increases strictly when a quasitransitive family of new edges is added. These results have the following implications for Cayley graphs: the connective constant of a Cayley graph decreases strictly when a new relator is added to the group, and increases strictly when a nontrivial group element is declared to be a generator.