On counting polygons in a crystal

How many $n$-step polygons exist that contain a given vertex of an infinite quasi-transitive graph $G$? The exponential growth rate of such polygons is identified as the connective constant when $G$ has sub-exponential growth and possesses a so-called square graph height function. The last condition amounts to the requirement that $G$ has a certain ${\mathbb{Z}}^2$ action of automorphisms. The main theorem extends a result of Hammersley (Proc. Cambridge Philos. Soc. 57 (1961) 516–523) and others for the hypercubic lattice, and responds to Hammersley’s challenge to prove such a result for more general "crystals’'.