Self-avoiding walks on cubic graphs and local transformations

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 $G_1=\phi (G)$, $$\mu (G{)}^{-1}=g(\mu (G_1{)}^{-1}),$$ 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.