The Cyclic Graph of a Finite Group

Abstract EN

The cyclic graph ΓG of a finite group G is as follows: take G as the vertices of ΓG and join two distinct vertices x and y if 〈x,y〉 is cyclic.

In this paper, we investigate how the graph theoretical properties of ΓG affect the group theoretical properties of G.

First, we consider some properties of ΓG and characterize certain finite groups whose cyclic graphs have some properties.

Then, we present some properties of the cyclic graphs of the dihedral groups D2n and the generalized quaternion groups Q4n for some n.

Finally, we present some parameters about the cyclic graphs of finite noncyclic groups of order up to 14.