Counting self-avoiding walks

TL;DR

This paper surveys how the connective constant of self-avoiding walks on graphs varies with graph transformations, bounds, and properties, highlighting open problems and implications for group theory.

Contribution

It provides a comprehensive overview of the dependence of the connective constant on graph structure, including new inequalities and effects of graph transformations.

Findings

Fisher transformation affects the connective constant on cubic graphs.

Bounds for the connective constant are established for regular graphs.

Strict inequalities for vertex-transitive graphs show how group modifications influence the connective constant.

Abstract

The connective constant $\mu(G)$ 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 $\mu(G)$ of the Fisher transformation (that is, the replacement of vertices by triangles). Secondly, we discuss upper and lower bounds for $\mu(G)$ when $G$ is regular. Thirdly, we present strict inequalities for the connective constants $\mu(G)$ 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.