The Cyclic Graph of a Finite Group

The cyclic graph of a finite group is as follows: take as the vertices of and join two distinct vertices and if is cyclic. In this paper, we investigate how the graph theoretical properties of affect the group theoretical properties of . First, we consider some properties of and characterize certain finite groups whose cyclic graphs have some properties. Then, we present some properties of the cyclic graphs of the dihedral groups and the generalized quaternion groups for some . Finally, we present some parameters about the cyclic graphs of finite noncyclic groups of order up to . 1. Introduction and Results Recently, study of algebraic structures by graphs associated with them gives rise to many interesting results. There are many papers on assigning a graph to a group and algebraic properties of group by using the associated graph; for instance, see [1–4]. Let be a group with identity element . One can associate a graph to in many different ways. Abdollahi and Hassanabadi introduced a graph (called the noncyclic graph of a group; see [4]) associated with a group by the cyclicity of subgroups. It is a graph whose vertex set is the set , where is cyclic for all and is adjacent if is not a cyclic subgroup. They established some graph theoretical properties (such as regularity) of this graph in terms of the group ones. In this paper, we consider the converse. We associate a graph with (called the cyclic graph of ) as follows: take as the vertices of and two distinct vertices and are adjacent if and only if is a cyclic subgroup of . For example, Figure 1 is the cyclic graph of , and Figure 2 is . For any group , it is easy to see that the cyclic graph is simple and undirected with no loops and multiple edges. By the definition, we shall explore how the graph theoretical properties of affect the group theoretical properties of . In particular, the structure of the group by some graph theoretical properties of the associated graph is determined. Figure 1: . Figure 2: . The outline of this paper is as follows. In Section 2, we introduce a lot of basic concepts and notations of group and graph theory which will be used in the sequel. In Section 3, we give some properties of the cyclic graph of a group on diameter, planarity, partition, clique number, and so forth and characterize a finite group whose cyclic graph is complete (planar, a star, regular, etc.). For example, the cyclic graph of any group is always connected whose diameter is at most 2 and the girth is either 3 or ; the cyclic graph of group is complete if and only if is cyclic and is a star if and