Cayley–Abels graph

Remarks

Example: let $G$ be a finitely generated group with the discrete topology, and let $S$ be a finite symmetric generating set. The Cayley graph of $(G, S)$, with $G$ acting by left multiplication, is connected because $S$ generates $G$. Each vertex has degree at most $\abs{S}$, the action is transitive, and stabilisers are trivial, hence compact and open.

Example: $G = \mathrm{Aut}(T_d)$, the automorphism group of the regular tree $T_d$ of degree $d \ge 3$, acting on $T_d$. Vertex stabilisers are compact and open in the topology of pointwise convergence.

Non-example: $\Z$ acting on its Cayley graph with respect to the infinite generating set $\Z$. The graph is complete, so every vertex has infinite degree, and clause 2 fails.

Construction: choose a compact open subgroup $U$ and a finite symmetric set $A$ with $G$ generated by $U \cup A$. Take the cosets $gU$ as vertices, and join $gU$ and $hU$ when $g^{-1} h \in U A U$. This is a Cayley–Abels graph, since $UAU$ is a finite union of cosets of $U$.