Uniform connective constant of a bounded-degree graph ($\mu ={\lim }_n({\sup }_v{\sigma }_n(v){)}^{1/n}$)

For an infinite connected locally finite graph with a uniform degree bound, let ${\sigma }_n(v)$ count its length-$n$ self-avoiding walks from $v$. The suprema satisfy ${\sigma }_{m+n}\le {\sigma }_m{\sigma }_n$, so the Fekete lemma applied to their logarithms gives a finite connective constant. Unlike the usual rooted definition on a vertex-transitive graph, this definition explicitly takes the supremum over roots.