  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. Firstly, the
connective constant decreases strictly when the graph is replaced by a
non-trivial quotient graph. Secondly, the connective constant increases
strictly when a quasi-transitive 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 non-trivial group element is declared to be a
generator.
