Self-avoiding walks and multiple context-free languages

Let $G$ be a quasi-transitive, locally finite, connected graph rooted at a vertex $o$, and let $c_n(o)$ be the number of self-avoiding walks of length $n$ on $G$ starting at $o$. We show that if $G$ has only thin ends, then the generating function $F_{\SAW,o}(z)=\sum_{n \geq 1} c_n(o) z^n$ is an algebraic function. In particular, the connective constant of such a graph is an algebraic number.

If $G$ is deterministically edge-labelled, that is, every (directed) edge carries a label such that no two edges starting at the same vertex have the same label, then the set of all words which can be read along the edges of self-avoiding walks starting at $o$ forms a language denoted by $L_{\SAW,o}$. Assume that the group of label-preserving graph automorphisms acts quasi-transitively. We show that $L_{\SAW,o}$ is a $k$-multiple context-free language if and only if the size of all ends of $G$ is at most $2k$. Applied to Cayley graphs of finitely generated groups this says that $L_{\SAW,o}$ is multiple context-free if and only if the group is virtually free.