Graphs admitting transitive commutative group actions

Let X be a compact metric space, and Homeo(X) be the group consisting of all homeomorphisms from X to X. A subgroup H of Homeo(X) is said to be transitive if there exists a point x∈X such that {k(x):k∈H} is dense in X. In this paper we show that, if X=G is a connected graph, then the following five conditions are equivalent: (1) Homeo(G) has a transitive commutative subgroup; (2) G admits a transitive Z2-action; (3) G admits an edge-transitive commutative group action; (4) G admits an edge-transitive Z2-action; (5) G is a circle, or a k-fold loop with k⩾2, or a k-fold polygon with k⩾2, or a k-fold complete bigraph with k⩾1. As a corollary of this result, we show that a finite connected simple graph whose automorphism group contains an edge-transitive commutative subgroup is either a cycle or a complete bigraph.