called an [**isomorphism**]{}. When $\gh_1 = \gh_2$, an isomorphism is called an [**automorphism**]{}. The group of all automorphisms of $\gh$ will be denoted by $\Aut(\gh)$. The action of a group $\gp$ on a graph $\gh$ by automorphisms is said to be [**transitive**]{} if there is only one $\gp$-orbit in $\verts(\gh)$ and to be [**quasi-transitive**]{} if there are only finitely many orbits in $\verts(\gh)$. A graph $\gh$ is [**transitive**]{} or [**quasi-transitive**]{} according as whether the corresponding action of $\Aut(\gh)$ is. For example, every Cayley graph is transitive. The action of a group on a set is called [**free**]{} if the stabilizer of each element of the set is just the identity element of the group. For example, every group acts on itself freely by multiplication.

A locally compact group is called [**unimodular**]{} if its left Haar measure is also right invariant. In particular, every discrete countable group is unimodular. We call a graph $\gh$ [**unimodular**]{} if $\Aut(\gh)$ is unimodular, where $\Aut(\gh)$ is given the weak topology generated by its action on $\gh$. Every Cayley graph and, as \[b\].SoardiWoess/ and \[b\].Salvatori/ proved, every quasi-transitive amenable graph is unimodular. See  for more details on unimodular graphs.

We now recall some definitions pertaining to 