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 Z 2 -action; (3) G admits an edge-transitive commutative group action; (4) G admits an edge-transitive Z 2 -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.