Non-cyclic Graph of a Group

We associate a graph ΓG to a non locally cyclic group G (called the non-cyclic graph of G) as follows: take G\Cyc(G) as vertex set, where Cyc(G) = {x ∈ G | 〈x, y〉 is cyclic for all y ∈ G}, and join two vertices if they do not generate a cyclic subgroup. We study the properties of this graph and we establish some graph theoretical properties (such as regularity) of this graph in terms of the group ones. We prove that the clique number of ΓG is finite if and only if ΓG has no infinite clique. We prove that if G is a finite nilpotent group and H is a group with ΓG ∼= ΓH and |Cyc(G)| = |Cyc(H)| = 1, then H is a finite nilpotent group. We give some examples of groups G whose non-cyclic graphs are “unique”, i.e., if ΓG ∼= ΓH for some group H , then G ∼= H . In view of these examples, we conjecture that every finite non-abelian simple group has a unique non-cyclic graph. Also we give some examples of finite non-cyclic groups G with the property that if ΓG ∼= ΓH for some group H , then |G| = |H |. These suggest the question whether the latter property holds for all finite non-cyclic groups.