Self-avoiding walks on cubic graphs and local transformations

TL;DR

This paper develops a general substitution principle for self-avoiding walks on cubic graphs, allowing the exact calculation of connective constants for new graphs via algebraic equations, extending known relations.

Contribution

It introduces a universal method to relate SAWs on original and transformed graphs through vertex replacements, generalizing the Fisher-triangle relation to arbitrary symmetric three-port gadgets.

Findings

Established a formula linking connective constants before and after vertex replacement.

Extended the Fisher-triangle relation to a broader class of graphs and gadgets.

Derived explicit algebraic equations for connective constants of transformed graphs.

Abstract

Despite its elementary definition, the self-avoiding walk (SAW) poses notoriously hard enumerative problems: exact connective constants are known for only a handful of infinite graphs, notably the honeycomb lattice \cite{ds}. We establish a general substitution principle for SAWs on infinite connected quasi-transitive cubic graphs under port-transitive vertex replacements, where each degree-$3$ vertex is replaced by a fixed finite three-port gadget. Writing $g(x)$ for the associated two-port SAW series, we prove that for $G_1=\phi(G)$, \[ \mu(G)^{-1}=g\bigl(\mu(G_1)^{-1}\bigr), \] equivalently $\mu(G_1)^{-1}$ is the unique solution $x\in(0,1)$ of $g(x)=\mu(G)^{-1}$, thereby extending the Fisher-triangle relation of Grimmett--Li to arbitrary symmetric three-port gadgets. We also obtain the corresponding identity for bipartite graphs when one or both colour classes are transformed, and show that the critical exponents $\gamma$ and $\eta$ (and $\nu$ under a standard regularity hypothesis) are invariant. For explicit gadget families, including complete-graph gadgets $K_N$ and Fisher-type constructions, these identities turn base graphs with known $\mu$ into infinite families of new quasi-transitive graphs whose connective constants are determined exactly as the unique roots of explicit algebraic equations.