Strict inequalities for connective constants of transitive graphs

TL;DR

This paper establishes strict inequalities for the connective constants of vertex-transitive graphs, showing how these constants change when graphs are quotiented or edges are added, with implications for Cayley graphs.

Contribution

It proves new strict inequalities for connective constants under graph quotients and edge additions, extending understanding of their behavior in transitive graphs.

Findings

Connective constant decreases when replacing a graph with a quotient

Connective constant increases when adding new edges

Results apply to Cayley graphs with relators and generators

Abstract

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.