Self-avoiding walks and multiple context-free languages

Let \(G\) be a quasi-transitive, locally finite, connected graph rooted at vertex \(o\), and let \(c_n(o)\) the number of self-avoiding walks length \(n\) on starting \(o\). We show that if has only thin ends, then generating function \(F_{\mathrm{SAW},o}(z)=\sum_{n \geq 1} c_n(o) z^n\) is an algebraic function. In particular, connective constant such number.If deterministically edge-labelled, is, every (directed) edge carries label no two edges same have label, set all words which can read along \(o\) forms language denoted by \(L_{\mathrm{SAW},o}\). Assume group label-preserving automorphisms acts quasi-transitively. \(L_{\mathrm{SAW},o}\) \(k\)-multiple context-free size ends most \(2k\). Applied to Cayley graphs finitely generated groups this says multiple virtually free.Mathematics Subject Classifications: 20F10, 68Q45, 05C25Keywords: Self avoiding walk, formal language, context free graph,