The Cyclic Graph of a Finite Group

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 groupsD 2n and the generalized quaternion groups Q 4n for some n. Finally, we present some parameters about the cyclic graphs of finite noncyclic groups of order up to 14.


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][2][3][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  \ Cyc(), where Cyc() = { ∈ |⟨, ⟩ 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 Γ  with  (called the cyclic graph of ) as follows: take  as the vertices of Γ  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  2 ×  2 , and Figure 2 is Γ  3 .For any group , it is easy to see that the cyclic graph Γ  is simple and undirected with no loops and multiple edges.By the definition, we shall explore how the graph theoretical properties of Γ  affect the group theoretical properties of .In particular, the structure of the group by some graph theoretical properties of the associated graph is determined.
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 Γ  of group  is complete if and only if  is cyclic and is a star if and only if  is an elementary abelian 2-group.In particular, for a finite group , Aut(Γ  ) = Aut() if and only if  ≅  2 ×  2 , the Klein group.In Section 4, we present some properties of the cyclic graphs of the dihedral groups  2 , including degrees of vertices, traversability (Eulerian and Hamiltonian), planarity, coloring, and the number of edges and cliques.Furthermore, we get the automorphism group of  2 for all  ≥ 3. Particularly, for all  > 2, if  is a group with Γ  ≅ Γ  2 , then  ≅  2 .Similar to Section 4, we discuss the properties of the cyclic graphs on the generalized quaternion groups  4 in Section 5.In Section 6, we obtain some parameters on the cyclic graphs of finite noncyclic groups of order up to 14.
In this paper, all groups considered are finite.Let  be a finite group with identity element .The number of elements of  is called its order and is denoted by ||.The order of an element  of  is the smallest positive integer  such that   = .The order of an element  is denoted by ||.For more notations and terminologies in group theory consult [6].

Some Properties of the Cyclic Graphs
Definition 1.In group theory, a locally cyclic group is a group in which every finitely generalized subgroup is cyclic.A group is locally cyclic if and only if every pair of elements in the group generates a cyclic group.It is a fact that every finitely generalized locally cyclic group is cyclic.So a finite locally cyclic group is cyclic.
Definition 2 (see [7,8]).Let  be a group.The cyclicizer of an element  of , denoted Cyc  (), is defined by In general, the cyclicizer Cyc  () of  is not a subgroup of .For example, let } is not a subgroup.
Definition 3. The cyclicizer Cyc() of  is defined as follows: By [9, Theorem 1], Cyc() is a normal subgroup of  and Cyc() ≤ ().Proof.By Definitions 2 and 4, it is straightforward.Proposition 6.Let  be a group with the identity element .Then diam(Γ  ) ≤ 2. In particular, Γ  is connected and the girth of Γ  is either 3 or ∞.
Proof.Suppose that  and  are two distinct vertices of Γ  .If ⟨, ⟩ is a cyclic subgroup of , then  is adjacent to , and hence (, ) = 1.Thus we may assume that ⟨, ⟩ is not cyclic.Note that both ⟨, ⟩ and ⟨, ⟩ are cyclic and the vertices  and  are adjacent to ; hence we get (, ) = 2.This means that Γ  is connected and diam(Γ  ) ≤ 2. If there exist  ̸ = ,  ̸ =  such that  and  are joined by some edge, then {, , } is a cycle of order 3 of Γ  and so the girth of Γ  is 3. Otherwise, every two vertices (nonidentity elements of ) of Γ  are not adjacent; that is, Γ  is a star, which implies that the girth of Γ  is equal to ∞.

Algebra 3
The following proposition is obvious; we omit its proof.Proof.Assume that Γ  is a star.Let  be a nonidentity element of .If || ≥ 3, then  and  −1 are adjacent since ⟨,  −1 ⟩ is a cyclic subgroup of , which is contrary to Γ  being a star.Hence || = 2.It follows that the order of every element of  is 2. If  and  are two elements of , then () 2 =  =  = , and hence  = .It means that  is an abelian group and exp() = 2.It follows that  is an elementary abelian 2-group.
Conversely, suppose that  is an elementary abelian 2group.Then the order of every cyclic subgroup of  is 2. Let  is a nonidentity element of .If there exists an element  such that ⟨, ⟩ is cyclic, then ⟨, ⟩ = ⟨⟩, which implies  ∈ ⟨⟩.Note that  is an element of order 2; then  =  as  ̸ = .It follows that the unique element  is adjacent to  in Γ  .So Γ  ≅  Proof.Assume, on the contrary, Cyc() ̸ = {}.Then there exist two adjacent vertices  and  such that ,  ∈ Cyc().Since || > 2, there is an element  such that  ̸ = ,  ̸ = .By Definition 3, {, , } is a cycle of length 3 and so the subgraph of Γ  induced by {, , } is an odd cycle, which is a contradiction to Γ  being bipartite (see [5,  Proof.Assume that deg Γ  () = || − 1 for every nonidentity element  of .If there exists an element  of  \ {} such that  is not of prime order, then we may choose  such that  divides the order of  and 1 <  < ||.Thus   ̸ =  and  is adjacent to   , since ⟨,   ⟩ = ⟨⟩.while  ∉ ⟨  ⟩ (otherwise, ⟨⟩ = ⟨  ⟩, a contradiction).so For the converse, suppose every nonidentity element  of  is of prime order.If ⟨, ⟩ is a cyclic subgroup of , then |⟨, ⟩| is a prime number.Thereby, ⟨, ⟩ = ⟨⟩, and so  ∈ ⟨⟩.That is, Cyc  () = ⟨⟩.Hence the theorem follows.
Theorem 23.Let  be a group.Then  Γ  () ∪ {} is a cyclic subgroup for all  ∈  \ {} if and only if every element  of  \ {} is contained in precisely one maximal cyclic subgroup of .
Theorem 24.Let  be a group.Then Aut(Γ  ) = Aut() if and only if  is isomorphic to the Klein group  2 ×  2 .
Proof.First we suppose that Aut(Γ  ) = Aut() for group .We shall show that  is isomorphic to the Klein group  2 × 2 by the following steps.
Step 3 ( is an elementary abelian 2-group).By Step 1, we have Obviously, there exists a graph automorphism  such that It follows that  is an elementary abelian 2-group.
Proposition 26.Let  be an elementary abelian -group for some prime integer .Then Γ  is isomorphic to Figure 3.
Proof.Let  be an element of  and  ̸ = .Since  is an elementary abelian -group, we conclude that the order of  is .It follows that the subgraph of Γ  induced by

Definition 4 .
Let  be a group.The cyclic graph Γ  of  is a graph with (Γ  ) =  and two distinct vertices ,  are adjacent in Γ  if and only if ⟨, ⟩ is a cyclic subgroup of .Proposition 5.For any group , deg Γ  () = |Cyc  ()| − 1, where  ∈ .

Proposition 7 .
Let  be a group with || > 2. Then {} is a dominating set of order 1 of Γ  .In particular, (Γ  ) = 1 and deg Γ  () = || − 1.Let  be a group.Then {} is a dominating set if and only if  ∈ Cyc().Moreover, the number of the dominating sets of size 1 is |Cyc()|.Let  be a nontrivial group.Then diam(Γ  ) = 1 (or equivalently Γ  is complete) if and only if  is a cyclic group.Proof.Let  and  be two arbitrary elements of .Suppose that diam(Γ  ) = 1.Then ⟨, ⟩ is a cyclic subgroup of .By Definition 1,  is a cyclic group as  is finite.For the converse, if  is a cyclic group, then ⟨, ⟩ is a cyclic subgroup of .Thus diam(Γ  ) = 1, as desired.Let  be noncyclic group.Then Γ  is not regular.Let  be a group with the identity element .Then Γ  ≅  1,||−1 (or equivalently Γ  is a star) if and only if  is an elementary abelian 2-group.