1 Introduction
Abstract

For a finite group with a normal subgroup , the enhanced quotient graph of , denoted by is the graph with vertex set and two vertices and are edge connected if or for some . In this article, we characterize the enhanced quotient graph of . The graph is complete if and only if is cyclic, and is Eulerian if and only if is odd. We show some relation between the graph and the enhanced power graph that was introduced by Sudip Bera and A.K. Bhuniya (2016). The graph is complete if and only if is cyclic if and only if is complete. The graph is Eulerian if and only if is odd if and only if is Eulerian, i.e., the property of being Eulerian does not depend on the normal subgroup .

Keywords: enhanced power graph; enhanced quotient graph; Eulerian graph; planar graph.
AMS Mathematics Subject Classification: 05C25, 05C38, 05C45.

The enhanced quotient graph of the quotient of a finite group

22footnotetext: This work was partially supported by CONACYT.

Luis A. Dupont, Daniel G. Mendoza and Miriam Rodríguez.

Circuito Gonzalo Aguirre Beltrán S/N;

Zona Universitaria;

Xalapa, Ver., México, CP 91090.

e-mail: ldupont@uv.mx

1 Introduction

The investigation of graphs related to groups as well as other algebraic structures is an exciting research topic in the last few decades,  [1, 2, 3, 4, 5, 6, 9, 11, 14, 16, 17, 18]. Only basic concepts about graphs will be needed for this paper. They can be found in any book about graph theory, for example  .
Given a finite group , there are many ways to associate a graph to by taking families of elements or subgroups as vertices and letting two vertices be joined by an edge if and only if they satisfy a property. All groups in this paper are finite. The enhanced power graph of a group was introduced by Sudip Bera and A. K. Bhuniya .

Definition 1.1.

Given a finite group , the enhanced power graph of denoted by is the graph with vertex set and two distinct vertices are edge connected in if there exists such that .

We will apply the idea of studying the enhanced power graph through the graphs of the quotient groups, as was carried out by the authors for the normal subgroup based power graph of a finite group in  .

Definition 1.2.

For a finite group with a normal subgroup , the enhanced quotient graph of , denoted by with vertex set and two vertices and are edge connected, (i.e. ), if or for some .

Let be a finite group of order and be a normal subgroup of with . In this paper we provide some results on the basic structure of the enhanced quotient graph and we show the interplay between the graph and the enhanced power graph .

2 Definitions and structure

In this section we provide the first results of the two graphs of groups that we will study in this article, the enhanced power graph of denoted by and the enhanced quotient graph of , denoted by . We will see the results of the graph in comparison to the graph which was introduced in .

is connected.

Proof.

Since and consequently is and edge of for all

Proposition 2.2.

For all is a clique of the graph

Proof.

Let and be elements of then hence

Corollary 2.3.

The graph contains at least isomorphic subgraphs to

Proposition 2.4.

If with and some element of forms an edge of some element with then is a clique of

Proof.

Suppose that with Then, for some that is, and for all and for all Therefore,

Corollary 2.5.

The graph contains at least isomorphic subgraphs to

Definition 2.6.

A clique , in a graph is a subset of the vertices of such that every two distinct vertices are adjacent. This is equivalent to the condition that the induced subgraph of induced by is a complete graph. A maximum clique of a graph is a clique such that there is no clique with more vertices. The clique number of a graph is the number of vertices in a maximum clique in .

Corollary 2.7.

For , , where is the highest order of a cyclic subgroup of .

Proof.

It follows from the proof of Proposition 2.4. ∎

Proposition 2.8.

Let be elements of Then, if and only if or

Proof.

If then for some Then, if we would have that On one hand, implies that for some thus On the other hand, implies that Hence,

Corollary 2.9.

The graph contains at least isomorphic subgraphs to

Definition 2.10.

For a finite group and an element of , we denote by the set of generators of .

Proposition 2.11.

 For with and we have that none of the vertices in is adjacent with vertices in in the graph of

Proposition 2.12.

For with and we have that none of the vertices in is adjacent with vertices in in the graph of and the graph

Proof.

For the graph it follows from Proposition 2.11. For we see that if and are adjacent in then for some Then with and a contradiction. ∎

Proposition 2.13.

 The graph contains a cycle if and only if for some

Proposition 2.14.

The graph contains a cycle if and only if for some

Proof.

Suppose that is a cycle with and for in Thus implies that there is such that Hence, The converse is trivial. ∎

Corollary 2.15.

Let be a group and let be a normal subgroup of Then the following conditions are equivalent.

1. is bipartite;

2. is bipartite;

3. is a tree;

4. is a tree;

5. is a star graph;

6. is a star graph.

3 Completeness

In this section we characterize when the graphs studied are complete graphs.

Theorem 3.1.

The graph is complete if and only if is cyclic.

Proof.

Suppose that Let vertices of Then, Thus, and therefore is complete. Conversely, suppose that is complete. Let us consider that with We claim that Let If is complete, then for some Thus, and By maximality and Therefore

Corollary 3.2.

 is complete if and only if is cyclic.

Take

Corollary 3.3.

is complete if and only if is cyclic if and only if is complete.

Definition 3.4.

Define a set as invertible if is closed under inverses and
The Cayley’s graph is defined by:

 VG,C:=V(C(G,C)):=G

where if and only if

We can summarize the theme of the graphs in the following result.

Theorem 3.5.

We have that

• where is the graph induced by in

• The following propositions are equivalent

1. is cyclic.

2. is -regular.

3. is complete.

4. is -regular.

4 Cone Property

Definition 4.1.

We say that a vertex of a graph is a cone vertex if is a edge of the graph for each vertex of A graph having a cone vertex is said to satisfy the cone property.

Remark 4.2.

For all finite groups we have that since one graph has more vertices than the other.

Theorem 4.3.

 Let be a finite group and . If , then has a cone vertex.

Theorem 4.4.

Let be a group and let be a normal subgroup of If with then each with is a cone vertex.

Proof.

The argument is similar to the proof of the Theorem 4.3. ∎

Theorem 4.5.

 Let be a finite abelian group. Then has a cone vertex if and only if has a cyclic Sylow -subgroup.

Theorem 4.6.

Let be a finite abelian group, with , and . Then has a cone vertex if and only if has a cyclic Sylow -subgroup.

Proof.

If then By Theorem 4.4 is a cone vertex.

Conversely, suppose that has not the cyclic Sylow -subgroup, we can assume that

Let be the cone vertex and consider with element of maximum order in . So if is a cone vertex, then which implies that for some -element By maximality, Then, with otherwise, it would be such that concluding that and similarly with implies that with a contradiction. ∎

Theorem 4.7.

 Let G be a non-abelian -âgroup. satisfies the cone property if and only if is generalized quaternion group.

Theorem 4.8.

Let be a normal subgroup of where is a -group and Then has the cone property if and only if is a generalized quaternion -group.

Proof.

Suppose that is a generalized quaternion -group. Thus, is the only subgroup of order Let then with Hence, Thus and therefore is a cone element. Conversely, let a cone element de for all with i.e., Therefore, Henceforth each element of of order belongs to a cyclic group. We conclude that has an unique subgroup of order

Theorem 4.9.

 Let be any simple group. Then does not satisfy the cone property.

Theorem 4.10.

Let be a maximal normal subgroup of . Then does not satisfy the cone property.

Proof.

Suppose that is a cone element, and is a prime such that Let with . Since , cyclic implies Therefore there exists a unique subgroup of order in the not simple group a contradiction. ∎

5 The properties: Eulerian, Hamiltonian and planar

A closed walk in a graph containing all the edges of is called an Euler path in . A graph containing an Euler path is called an Euler graph or Eulerian graph. The following theorem due to Euler , characterises Eulerian graphs. Euler proved the necessity part and the sufficiency part was proved by Hierholzer, .

Theorem 5.1.

(Euler) A connected graph is an Euler graph if and only if all vertices of are of even degree.

Theorem 5.2.

 Let be a group of order . Then the enhanced power graph is Eulerian if and only if is odd.

Theorem 5.3.

The graph is Eulerian if and only if is odd.

Proof.

If is odd, then is Eulerian. If are the maximal cyclic subgroups of that contain , with . Then

 deg(g)=[(|C1|−1)|H|+1]+s∑i=2|¯¯¯¯¯¯Ci−Ai||H|,

where is the maximum of that is contained in some , for . Thus, is even and is even.

Suppose that is Eulerian, consequently is even. Therefore and are both even or they are both odd.

Suppose that and are both even, then for

 deg(g)=[(|C1|−1)|H|+1]+s∑i=2|¯¯¯¯¯¯C2−A2||H|

where and are both even. Therefore, the degree of is odd, a contradiction. ∎

We conclude that the property of being Eulerian does not depend on the normal subgroup .

Corollary 5.4.

is Eulerian if and only if is odd if and only if is Eulerian.

Definition 5.5.

A graph is called Hamiltonian if it has a cycle that meets every vertex. Such a cycle is called a Hamiltonian cycle.

Theorem 5.6.

Let be a finite group and be a normal subgroup. Then the enhanced quotient graph is Hamiltonian if the enhanced power graph is Hamiltonian.

Proof.

Let the enhanced power graph be Hamiltonian. Suppose . Since is Hamiltonian, there exists a Hamiltonian cycle . So, implies that for all and every coset is a clique in . Hence, we can construct a Hamiltonian cycle in as follows:

 e∼a1h1∼a1h2∼⋯∼a1hm
 a1hm∼a2h1∼a2h2∼⋯∼a2hm
 a2hm∼⋯∼anh1∼anh2∼⋯∼anhm∼e.

Definition 5.7.

A planar graph is a graph that can be embedded in the plane, i.e., it can be drawn on the plane in such a way that its edges intersect only at their endpoints.

Theorem 5.8.

(Kuratowski 1930,  ) A finite graph is planar if and only if it does not contain a subgraph that is homeomorphic to or .

Theorem 5.9.

 Let a group. Then the enhanced power graph is planar if and only if

Theorem 5.10.

The graph is not planar if and only if or and or and or and , where is the highest order of a cyclic subgroup of .

Proof.

It follows from the Proposition 2.4 and Corollary 2.7. ∎

6 Deleted enhanced quotient graph

Definition 6.1.

In this section we consider the graphs obtained by deleting the vertex from the graphs , respectively. We call the deleted enhanced power graph and the deleted enhanced quotient graph. The deleted graphs are not necessarily connected. .

Theorem 6.2.

 Let be a finite –group. Then is connected if and only if has a unique minimal subgroup.

Theorem 6.3.

Let be a finite –group. Then is connected if and only if has a unique minimal subgroup.

Proof.

The argument is similar to the proof of the Theorem 6.2. ∎

For a finite group G, you have the following definitions:

 Πϵ(G)={|x|:x∈G},Π(G)={p∈N:p∣|G|\textmdisaprime},

and let be the set of all maximal element of under the divisibility relation.

Theorem 6.4.

 Let . Then the graph is connected.

Theorem 6.5.

For with , is connected.

Proof.

If is abelian then We consider such that We claim that for each

• implies that there are and prime numbers such that and
and implies that
Therefore

• Thus there is or

• Let be primes with with a similar argument we have that The other subcase is similar.

A last result of the same nature, concerning the deleted enhanced power graph which has its analogue for the deleted enhanced quotient graph is the following:

Theorem 6.6.

 Let and for some prime . Then the graph is connected if and only if for some non–central element of order there exists a non –element such that in the graph .

Finally, the Corollary 2.15 for deleted graphs can be rewritten as follows:

Theorem 6.7.

Let be a group. Then the following conditions are equivalent.

1. is bipartite;

2. is a forest;

3. has no cycle;

4. for every .

The circumference of a graph is the length of the longest cycle.

Theorem 6.8.

Let be a group and let be a normal subgroup of , with . Then the following conditions are equivalent.

1. is -partite with partition ; where for all .

2. The circumference of is ;

3. The clique number ) of is ;

4. for every ;

5. .

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