# Self-avoiding walks and multiple context-free languages

@article{Lehner2020SelfavoidingWA, title={Self-avoiding walks and multiple context-free languages}, author={Florian Lehner and Christian Lindorfer}, journal={ArXiv}, year={2020}, volume={abs/2010.06974} }

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 $F_{\mathrm{SAW},o}(z)=\sum_{n \geq 0} c_n(o) z^n$ 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…

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