Counting self-avoiding walks on free products of graphs

TL;DR

This paper derives a formula for the connective constant of self-avoiding walks on free products of graphs, showing explicit calculations for finite products and establishing algebraic properties of the constant.

Contribution

It provides a new formula for the connective constant of free products of graphs and demonstrates explicit computation and algebraic nature for finite cases.

Findings

Explicit formula for the connective constant of free products

Asymptotic growth rate of self-avoiding walks established

Connective constant is algebraic for finite products

Abstract

The connective constant $\mu(G)$ of a graph $G$ is the asymptotic growth rate of the number $\sigma_{n}$ of self-avoiding walks of length $n$ in $G$ from a given vertex. We prove a formula for the connective constant for free products of quasi-transitive graphs and show that $\sigma_{n}\sim A_{G} \mu(G)^{n}$ for some constant $A_{G}$ that depends on $G$. In the case of finite products $\mu(G)$ can be calculated explicitly and is shown to be an algebraic number.