The Cyclic Graph of a Finite Group

Abstract

The cyclic graph Γ G of a finite group G is as follows: take G as the vertices of Γ G and join two distinct vertices x and y if 〈 x , y 〉 is cyclic. In this paper, we investigate how the graph theoretical properties of Γ G affect the group theoretical properties of G . First, we consider some properties of Γ G and characterize certain finite groups whose cyclic graphs have some properties. Then, we present some properties of the cyclic graphs of the dihedral groups D 2 n and the generalized quaternion groups Q 4 n for some n . Finally, we present some parameters about the cyclic graphs of finite noncyclic groups of order up to 14 .