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\begin{document}
	\nocite{*}
	
\title{Intersection Normal Graphs of Finite Groups}
\author[$\star$]{H. M. Mohammed Salih}
\author[$\dagger$]{S. M. S. Omer}

\affil[$\star$]{Department of Mathematics, faculty of Science, Soran University, Kawa St. Soran, Iraq.}
\affil[$\dagger$]{Department of Mathematics, faculty of Science, University of Benghazi, Al Kufra, Libya.}

	\maketitle
\begin{abstract}
	In this paper, a new type of graph on a finite group $G$, namely the intersection normal graph is defined and studied. The graph is denoted by $\Gamma^{in}_G(N)$, where its vertices are normal subgroups of $G$, in which two distinct vertices $N_1$ and $N_2$ are adjacent if $N_1\cap N_2\subseteq N$ for a fixed normal subgroup $N$ of $G$. In this paper, the intersection graph is shown as a simple connected with a diameter less than or equal to two. Several graph properties are considered. However the graph structure of $\Gamma^{in}_G(\{e\})$ is given for some finite groups such as the dihedral, quaternion and cyclic groups.
	
	\let\thefootnote\relax\footnotetext{Received: xx xxxxx 20xx,\quad
		Accepted: xx xxxxx 20xx.\\[3ex]
	}
	
\end{abstract}


	% Separate keyword by \sep
\textbf{Keyword:normal subgroups; planer graph; cut-vertex}. 
	
	% Write the classification number
	Mathematics Subject Classification : 20P05, 20B40, 97K30.
	




%% Main text
\section{Introduction}

%Graph theory is the study of vertices and edges. More precisely, it involves the ways in which sets of points can be connected by edges. The concept in graph theory is widely used among many fields and one of these uses are in group theory. 
%In this section, some basic definitions that are needed in this paper are stated, starting with some definitions related to graph theory that can be found in one of the references (\cite{B1}and \cite{C1}). 
%A graph $\Gamma$ is a mathematical structure consisting of two sets namely vertices and edges which are denoted by $V(\Gamma)$ and $E(\Gamma)$, respectively.
%
%A graph is called directed if its edges are identified with ordered pair of vertices. Otherwise, $\Gamma$ is called indirected. Two vertices  are adjacent if they are linked by an edge. A connected graph is a graph in which there is a partition of vertex $V$ into non empty subsets, $V_1, V_2, ..., V_n$ such that two vertices $\omega_1$ and $\omega_2$ are connected if and only if they belong to the same set $V_i$. Subgraphs $\Gamma(V_1), \Gamma(V_2), ..., \Gamma(V_n)$ are all components of $\Gamma$. The graph $\Gamma$ is connected, if it has precisely one component. Thus, a subgraph of a graph $\Gamma$ is a graph whose vertices and edges are subset of the vertices and edges of $\Gamma$. Hence we denote $\Gamma_{\text{sub}}$ a subgraph of $\Gamma$. 
%However, a complete graph is a graph where each ordered pair of distinct $n$ vertices are adjacent, and it is denoted by $K_n$.
%

%This paper is divided into three sections. The first section focuses on some background topics on graph theory and algebra, while the second section provides some earlier and recent publications that are related to the probability that a group element fixes a set and some graphs. In the third section, we present our main results on which include the orbit graph.

A group theory can be considered as the study of symmetry. A group is basically the collection of symmetries of some object preserving some of its structure; therefore many mathematicians could asscoited the group theory with graph theory such as \cite{erfanian2012conjugate},\cite{sarmin2017graphs} and \cite{tong}. It has been proved that graphs can be interesting tools for the study of groups. Groups linked with graphs have been arguably the most famous and productive area of algebraic graph theory. 

In the following context, some basics and related works are provided. 

A graph is connected if there is a path connecting any two distinct vertices. The distance between two distinct vertices is the length of the shortest path connecting them (if such a path does not exist, define $\infty$. The diameter of a graph $G$, denoted by $diam(G)$, is defined by the supremum of the distances between vertices. The girth of a graph, denoted $g(G)$ is the length of the shortest cycle in the graph $G$. A graph with no cycles has infinite girth. The minimum among all the maximum distances between a vertex to all other vertices is considered as the radius of the graph $G$ and denoted by $rad(G)$.

The $r$-partite graph is one whose vertex can be partitioned into $r$ subsets so that an edge has both ends in no subset. A complete $r$-partite graph is an $r$-partite graph in which each vertex is adjacent to every vertex that is not in the same subset. The complete bipartite graph with part sizes $m$ and $n$ is denoted by $K_{m,n}$. A graph is called a complete if each pair of vertices is joined by an edge. We use $K_n$ to denote the complete graph with $n$ vertices. 
Two graphs $G$ and $H$ are isomorphic, denoted by $G\cong H$, if there is
a bijection $\phi\colon G\rightarrow H$ of vertices such that the vertices $x$ and $y$ are adjacent in $G$ if and only if $\phi(x)$ and $\phi(y)$ are adjacent in $H$. A connected graph can be drawn without any edges crossing, it is called planar. A vertex $v$ of a connected graph $G$ is called a cut vertex of $G$, if $G\setminus v$ (Delete $v$ from $G$) results in a disconnected graph. Removing a cut vertex from a graph breaks it into two or more graphs \cite{wilson1979introduction},\cite{ye2012co}. A non empty subset $S$ of a group $H$ is called subgroup if $S$ is a group and denoted by $S\leq H$. A subgroup $S$ of $H$ is called normal if $h^{-1}sh\in S$ for all $h\in H$ and $s\in S$ \cite{kurzweil2006theory}.

\begin{theorem}\cite{kurzweil2006theory}
\label{t1}
Let $N$ be a minimal normal subgroup of $G$. For all normal subgroups $M$ of $G$ either $N \leq  M$ or $N\cap M =1$.
\end{theorem}

\begin{theorem}[Kuratowski's Theorem]\cite{wilson1979introduction} A graph is non-planar if and only if it contains a subgraph homeomorphic to $K_{3,3}$ or $K_5$.
\label{pp11}
\end{theorem}

\section{Main Results}
In this section, a new graph namely the intersection normal graph is introduced. Besides, some related results are obtained.

\begin{definition}
Let $G$ be a finite group and $\mathcal{N}(G)$ be the set of all normal subgroups of $G$ and $N\in \mathcal{N}(G)$. The intersection normal graph denoted by $\Gamma_{G}^{in}(N)$ is a undirected graph whose vertex set is $\mathcal{N}(G)$ and two distinct vertices $N_i$ and $N_j$ are adjacent if $N_i\cap N_j\subseteq N$.
\label{d1}
\end{definition} 

\begin{example}
Consider the group of integer modulo 4, that is $\mathbb{Z}_4$. The interaction normal graphs for normal subgroups $\{0\},\{0,2\}$ and $\mathbb{Z}_4$ are $\Gamma_{G}^{in}(\{0\})=K_{1,2}$ and $\Gamma_{G}^{in}(\{0,2\})=\Gamma_{G}^{in}(\mathbb{Z}_4)=K_3$. 
\end{example}


\begin{example}
Consider the Klein four group $V_4=\{e,a,b,c\}$ where $a^2=b^2=c^2=e$ and $ab=ba, ac=ca,bc=ca$. The interaction normal graph $\Gamma_{G}^{in}(\{e\})$ is given in Figure \ref{ff2}.

\newpage

\begin{figure}
	\centering
	\begin{tikzpicture}
	\node[circle, fill=blue,inner sep=0pt,label=above:{$<a>$}, minimum size=5pt] (a) at (-2,1) {};
	\node[circle, fill=red,inner sep=0pt,label=above:{$<b>$}, minimum size=5pt] (b) at (0,0) {};
	\node[circle, fill=green,inner sep=0pt,label=above:{$<c>$}, minimum size=5pt] (c) at (-4,0) {};
	\node[circle, fill=green,inner sep=0pt,label=below:{$V_4$}, minimum size=5pt] (d) at (-2,-6) {};
	\node[circle,fill=yellow,inner sep=0pt,minimum size=5pt,label=right:{$<e>$}] (e) at (-2,-3) {};
	
	\draw (a)--(b)--(c)--(e);
	\draw (c)--(a)--(e);
	\draw (b)--(e);
	\draw (d)--(e);
	\draw (a)--(c);
	\end{tikzpicture}
	\caption{The intersection normal graph of $\{e\}$}
	\label{ff2}	
\end{figure}

\end{example}



\begin{remark}
A finite group $G$ is simple if and only if $\Gamma_{G}^{in}(N)=K_{2}$.
\label{t55}
\end{remark}
%\begin{proof}
%Since $G$ is a finite simple group, then it has only two normal subgroups which are $\{e\}$ and $G$. Now $\{e\}\cap G\subseteq N$ where $N\in \{ \{e\},G\}$. So $\{e\}$ and $G$ are adjacent. Thus $\Gamma_{G}^{in}(N)=K_{2}$.
%Conversely, let $\Gamma_{G}^{in}(N)=K_{2}$, then $G$ has only two normal subgroups and it must be $\{e\}$ and $G$. The proof then follows. \end{proof}


\begin{lemma}
Let $N_1,N_2,...,N_l$ be normal subgroups of a finite group $G$. Then 
\begin{enumerate}
	\item $\Gamma_{G}^{in}(G)=K_{|\mathcal{N}(G)|}$.
	\item $\Gamma_{G}^{in}(\bigcap_{i\in \Lambda} N_i)=\bigcap_{i\in \Lambda} \Gamma_{G}^{in}(N_i)$.
\end{enumerate}
\end{lemma}

\begin{proof}
The proof is clear.
\end{proof}

\begin{lemma}
Let $N$ be a minimal normal subgroup of $G$ and $L$ be a non trivial normal subroup of $G$. Then $deg_{\Gamma_{G}^{in}(L)}(\{e\})=deg_{\Gamma_{G}^{in}(L)}(N)=|\mathcal{N}(G)|$.
\label{l2}
\end{lemma}
\begin{proof}
Based on Theorem \ref{t1}, thus $N\cap L=N$ or $N\cap L=\{e\}$. In both cases we obtain $N\subseteq L$ or $\{e\}\subset L$. Thus $deg_{\Gamma_{G}^{in}(L)}(N)=|\mathcal{N}(G)|$. \end{proof}

\begin{proposition}
If $G$ has normal subgroups $N_i$ such that $\{e\}=N_0\subset N_1\subset N_2\subset ...\subset N_r\subset N_{r+1}=G$, then 
\begin{enumerate}
	\item $\Gamma_{G}^{in}(G)$ and $\Gamma_{G}^{in}(N_r)$ are identical.
	\item $\Gamma_{G}^{in}(\{e\})=K_{1,r+1}$
	\item $\Gamma_{G}^{in}(N_i)$ is a subgraph of $\Gamma_{G}^{in}(N_{l})$ where $l>i$.
\end{enumerate}
\label{p11}
\end{proposition}

\begin{proof}
%	\begin{enumerate}
First, it is clear that they have the same number of vertices. Let $e=N_iN_j$ be an edge in $\Gamma_{G}^{in}(N_r)$, that is $N_i\cap N_j\subseteq N_r\subseteq G$. This implies that $e=N_iN_j$ is an edge in $\Gamma_{G}^{in}(G)$. On the other hand, let $e=N_iN_j$ be an edge in $\Gamma_{G}^{in}(G)$, that is $N_i\cap N_j\subseteq G$. If either $N_i$ or $N_j$ is $N_r$, then we are done. If neither $N_i$ nor $N_j$ is $N_r$, then $N_i\cap N_j= N_{\bar{r}}\subseteq N_r$ where $\bar{r}=min\{i,j\}$. Thus $e$ is an edge of $\Gamma_{G}^{in}(G)$. 
Second, Since $N_i\cap N_j=N_l\not\subset \{e\}$ for $i,j\in \{1,2,...,r+1\}$ where $l=min \{i,j\}$ and $N_i\cap \{e\}=\{e\}$, then the result follows. Third, the proof is clear.
%\end{enumerate}
\end{proof}

As a direct consequence of Proposition \ref{p11}, the following results are obtained.
\begin{corollary}
If $G$ has a subnormal series $\{e\}\triangleleft N_1\triangleleft ...\triangleleft N_r\triangleleft G$.
\begin{enumerate}
	\item If it has length 3, then $\Gamma_{G}^{in}(\{e\})\subseteq \Gamma_{G}^{in}(N_1)\subseteq\Gamma_{G}^{in}(G)$ or $( K_{1,2}\subseteq K_3\subseteq K_3)$.
	\item If it has length 4, then $\Gamma_{G}^{in}(\{e\})\subseteq \Gamma_{G}^{in}(N_1)\subseteq \Gamma_{G}^{in}(N_2) \subseteq \Gamma_{G}^{in}(G)$ 
	
	or $( K_{1,3}\subseteq K_4\setminus\{one \:edge\} \subseteq K_4\subseteq K_4)$.
\end{enumerate}
\end{corollary}


\begin{proposition}
If $G$ is a finite solvable group and $\Gamma_{G}^{in}(\{e\})=K_{1,2}$, then 
\begin{enumerate}
	\item $G$ is a cyclic $p$-group of order $p^2$.
	\item $G$ is a semidirect product $G = P\rtimes Q$, where $P$ is an elementary abelian $p$-group and $Q$ is a cyclic group of order $q$, with $p$ and $q$ being distinct primes. Moreover, the action of $Q$ on $P$
	is irreducible.
\end{enumerate}
\label{pp1}
\end{proposition}
\begin{proof}
Since $\Gamma_{G}^{in}(\{e\})=K_{1,2}$, then $G$ has a unique non trivial normal subgroup. The proof of rests follow from Theorem 1.2 in \cite{hai2008finite}.
\end{proof}

Note that the converse of 1. in Proposition \ref{pp1} is true and it can be seen in Proposition \ref{j1}. The following example shows that the converse of 2. in Proposition \ref{pp1} is not true. 


\begin{example}
Any group of order 20 has a normal 5-Sylow subgroup and is semidirect product. But $\Gamma_{G}^{in}(\{e\})\neq K_{1,2}$.
\end{example}


\begin{proposition}
If $G$ is a finite non solvable group and $\Gamma_{G}^{in}(\{e\})=K_{1,2}$, then $G$ describe in Theorem 1.3 in \cite{hai2008finite}.
\end{proposition}
\begin{proof}
The proof is similar as Proposition \ref{pp1}.
\end{proof}


\begin{example}
Consider the group $G=GL(2,3)$ and it has 5 normal subgroups such as $\{e\},C_2,Q_8, SL(2,3)$ and $GL(2,3)$. The corresponding intersection normal graphs are $K_{1,4}\subseteq H\subseteq K_4\setminus\{e\}\subseteq K_4\subseteq K_4$ where the intersection normal graph of $H$ is given in Figure \ref{ff1}.

\newpage
\begin{figure}
	\centering
	\begin{tikzpicture}
	\node[circle, fill=blue,inner sep=0pt,label=above:{$C_2$}, minimum size=5pt] (a) at (-2,-1) {};
	\node[circle, fill=red,inner sep=0pt,label=above:{$GL(2,3)$}, minimum size=5pt] (b) at (0,0) {};
	\node[circle, fill=green,inner sep=0pt,label=above:{$SL(2,3)$}, minimum size=5pt] (c) at (-4,0) {};
	\node[circle, fill=green,inner sep=0pt,label=below:{$Q_8$}, minimum size=5pt] (d) at (-2,-6) {};
	\node[circle,fill=yellow,inner sep=0pt,minimum size=5pt,label=below:{$<e>$}] (e) at (-2,-3) {};
	
	\draw (c)--(b)--(d);
	\draw (a)--(b);
	\draw (b)--(e);
	\draw (e)--(c)--(d);
	\draw (a)--(c);
	\end{tikzpicture}
	\caption{The intersection normal graph of $H$ }	
	\label{ff1}
\end{figure}

\end{example}

\begin{lemma}
Let $\Gamma_{G}^{in}(N)$ be the intersection normal graph. Then 	
\begin{displaymath}
d(H_i,H_j)= \left\{
\begin{array}{lr}
1 &   if \:  H_i\cap H_j\subseteq N  \\
2 &   if \:  H_i\cap H_j\not\subseteq N  \\
\end{array}
\right.
\end{displaymath}
\label{l1} \end{lemma}

\begin{proof}
The proof is clear. \end{proof}

\begin{proposition}
Let $G$ be a finite group and $N$ be a normal subgroup of $G$. Then $\Gamma_{G}^{in}(N)$ is connected graph with diameter at most 2 and radius 1. \end{proposition}

\begin{proof}
%Let $N_i$ and $N_j$ be two vertices in $\Gamma_{G}^{in}(N)$. If $N_i\cap N_j\subseteq N$, then $N_i$ and $N_j$ are adjacent. If $N_i\cap N_j\not \subseteq N$, then $N_i\cap \{e\}=\{e\}$ and $N_j\cap \{e\}=\{e\}$. This implies that $N_i$ is adjacent to $\{e\}$ and $\{e\}$ is adjacent to $N_j$. So there is a path from $N_i$ to $N_j$. Thus $\Gamma_{G}^{in}(N)$ is connected graph. The proof of the rest follows Lemma \ref{l1}. 
The proof is clear.
\end{proof}



\begin{proposition}
If $\Gamma_{G}^{in}(\{e\})$ contains a cycle, then $\Gamma_{G}^{in}(N)$ has girth 3.
\end{proposition}
\begin{proof}
Since $\Gamma_{G}^{in}(\{e\})$ contains a cycle, then there are normal subgroups $N_1$ and $N_2$ of $G$ such that $N_1\cap N_2=\{e\}$. These normal subgroups together with $\{e\}$ give $K_3$. Based on Proposition \ref{p11}, $\Gamma_{G}^{in}(\{e\})$ is a subgraph of $\Gamma_{G}^{in}(N)$. Hence $\Gamma_{G}^{in}(N)$ has girth 3.
\end{proof}

\begin{proposition}
Let $G$ be a finite group with at least two minimal normal subgroups. Then $\Gamma_{G}^{in}(\{e\})$ is not tree graph.
\label{p1}
\end{proposition}
\begin{proof}
Since $G$ has at least two minimal normal subgroups $N_1$ and $N_2$, thus the normal subgroups with trivial normal subgroup gives the cycle $K_3$ in $\Gamma_{G}^{in}(\{e\})$. Therefore $\Gamma_{G}^{in}(\{e\})$ is not tree.\end{proof}


\begin{theorem}
Let $G$ be a finite group with non trivial normal proper subgroups $N_1,...,N_r$. If $|E(\Gamma_{G}^{in}(N_i))|=|E(\Gamma_{G}^{in}(N_j))|$ and $|N_i|=|N_j|$ for some $i,j$. Then $\Gamma_{G}^{in}(N_i)$ and $ \Gamma_{G}^{in}(N_j)$ are isomorphic.
\label{t44}
\end{theorem}
\begin{proof}
Define $f\colon V(\Gamma_{G}^{in}(N_i))\rightarrow V(\Gamma_{G}^{in}(N_j))$ by $f(H_i)=H_j$, $f(H_j)=H_i$ and $f(H_l)=H_l$ where $l\neq i,j$. It is clear that $f$ is bijective. Let $H_mH_s$ be an edge in $\Gamma_{G}^{in}(N_i)$ that is $H_m\cap H_s\subseteq H_i$. If $m,s\neq i,j$ then $f(H_m\cap H_s)\subseteq f(H_i)$, that is $H_m\cap H_s\subseteq H_j$. Thus, $H_mH_s$ is an edge in $\Gamma_{G}^{in}(N_j)$. If $m=i$ or $s=i$ $(m=j\text{ or } s=j)$, then the result follows.
\end{proof}

The following examples show that the converse of Theorem \ref{t44} is not true.
\begin{example}
Let $G=C_{15}$ be a cyclic group. Then, $\Gamma_{G}^{in}(C_3) \cong \Gamma_{G}^{in}(C_5)$. 
\end{example}

\begin{example}
The groups $D_8$ and $Q_8$ have the same number of normal subgroups. Thus, their intersection normal graphs are identical.
\end{example}

\begin{proposition}
Let $G$ be a finite group with at least four minimal normal subgroups. Then $\Gamma_{G}^{in}(N)$ is not planar graph.
\end{proposition}
\begin{proof}
Suppose that $G$ is finite group with four minimal normal subgroups $N_i$ for $i=1,2,3,4$. It is clear that $N_i\cap N_j=\{e\}$ for $i\neq j$. These minimal normal subgroups together with trivial normal group produce $K_5$ in $\Gamma_{G}^{in}(\{e\})$. From Theorem \ref{pp11}, the result is obtain. \end{proof}

\begin{proposition}
Let $\Gamma_{G}^{in}(\{e\})$ be the intersection normal graph, with at least two minimal normal subgroups of $G$ then $\{e\}$ is a cut vertex.
\end{proposition}
\begin{proof}
If $\Gamma_{G}^{in}(\{e\})$ is a star graph, then the proof is clear. 
If $\Gamma_{G}^{in}(\{e\})$ is not star graph, then using Lemma \ref{l2}, $deg_{\Gamma_{G}^{in}(\{e\})}(\{e\})=l$ where $l=|\mathcal{N}(G)|$ and $deg_{\Gamma_{G}^{in}(\{e\})}(G)=1$. Therefore, vertex $G$ is isolated vertex in $\Gamma_{G}^{in}(\{e\})\setminus\{e\}$. Thus, the graph is disconnected and $\{e\}$ is a cut vertex. \end{proof}


\begin{proposition}\label{t4}
Let $G\cong \langle a,b |a^n=e=b^2, bab^{-1}=a^{-1}\rangle$ be the dihedral group $D_{2n}$. Then

\begin{equation*}
\Gamma_{G}^{in}(\{e\})=
\begin{cases}
K_{1,\alpha+1}, \text{if $n$ is odd, and $n=p^{\alpha}$},\\
K_{1,\alpha+3}, \text{if $n$ is even, and $n=p^{\alpha}$},\\
\text{ is not tree}, \text{if there exist distinct primes $p_i$ and $p_j$} \\ \text{such that $n=p_1^{\alpha_1}...p_r^{\alpha_r}$}.
\end{cases}
\end{equation*}

%	\begin{displaymath}
%	\Gamma_{G}^{in}(\{e\})= \left\{
%	\begin{array}{lr}
%	K_{1,\alpha+1};   if \: n \text{ is odd and } n=p^{\alpha}  \\
%	K_{1,\alpha+3};   if \: n \text{ is even and } n=p^{\alpha}  \\
%	\text{ is not tree};  \: if \text{there exist distinct primes} p_i \text{ and }p_j\\ \text{ such that } n=p_1^{\alpha_1}...p_r^{\alpha_r}. \\
%	\end{array}
%	\right.
%	\end{displaymath}
\end{proposition}
\begin{proof}
%\begin{enumerate}
When $n$ is odd, it is clear that $G$ has $\alpha+2$ normal subgroups. The rest follows from Proposition \ref{p11}. In the case that $n$ is even, thus $G$ has $\alpha+4$ normal subgroups. Recall Proposition \ref{p11}, then $K_{1,\alpha+3}$. Since $n=p_1^{\alpha_1}...p_r^{\alpha_r}$, then without loss of generality we assume that $n=p_1^{\alpha_1}p_2^{\alpha_2}$. The proof of the rest follows from Proposition \ref{p1}.
%	\end{enumerate}

\end{proof}

\begin{proposition}
Let $G=Q_{4n}$ be a quaternion group. Then
\begin{equation*}
\Gamma_{G}^{in}(\{e\})=
\begin{cases}
K_{1,\alpha+4},  \text{if $n=2^{\alpha}$},  \\
\text{ is not tree,  otherwise} \\
\end{cases}
\end{equation*}

\end{proposition}
\begin{proof}
The proof is similar as Proposition \ref{t4}. \end{proof}


\begin{proposition}
If $G$ is a cyclic group of order $p^n$ where $p$ is a prime number, then $\Gamma_{G}^{in}(\{e\})$ is a star graph $(\Gamma_{G}^{in}(\{e\})=K_{1,n})$.
\label{j1}	
\end{proposition}
\begin{proof}
If $n=1$, then $G$ is simple group. From Proposition \ref{t55}, we have  $\Gamma_{G}^{in}(\{e\})=K_{1,1}$ and the proof of $n>1$ follows from Proposition \ref{p11}.\end{proof}

\begin{proposition}
If $G$ is a finite abelian group such that $G$ is not cyclic, then $\Gamma_{G}^{in}(\{e\})$ is not a tree graph.
\end{proposition}
\begin{proof}
The proof is clear.\end{proof}

\section{Conclusion}
In this paper, we introduced a new graph called the intersection graph whose vertices are normal subgroups. The graph is found for dihedral groups, quaternion groups and others. Besides, some properties of the intersection graph were determined.


%\section*{Acknowledgement} 
%Write support, acknowledgment, dedicatory, and grants here.

%\section*{References} 
\begin{thebibliography}{99}
\bibitem{tong}
Tong-Viet, Hung P, Finite groups whose prime graphs are regular \textit{Journal of Algebra,} \textbf{397}, (2014), 18-31.

\bibitem{kurzweil2006theory}
Kurzweil, Hans and Stellmacher, Bernd, The theory of finite groups: an introduction, \textit{Springer Science \& Business Media}, (2006).

\bibitem{wilson1979introduction}
Wilson, Robin J, Introduction to graph theory, \textit{Pearson Education India}, (1979).

%\bibitem{Miller}
%G.A. Miller, Relative Number of Non-invariant Operators in a Group, {\it Proc. Nat. Acad. Sci.}, USA, {\bf 30}(20), (1994), 25-28.


\bibitem{erfanian2012conjugate}
Erfanian, Ahmad and Tolue, Behnaz, Conjugate graphs of finite groups, \textit{ Discrete Mathematics, Algorithms and Applications.}, \textbf{4}(02) (2012).
%
\bibitem{sarmin2017graphs}
Sarmin, Nor Haniza and Noor, Alia Husna Mohd and Omer, Sanaa Mohamed Saleh, ON GRAPHS ASSOCIATED TO CONJUGACY CLASSES OF SOME THREE-GENERATOR GROUPS, \textit{JURNAL TEKNOLOGI.}, \textbf{79}(1) (2017), 55-61.

\bibitem{ye2012co}
Ye, Meng and Wu, Tongsuo, Co-maximal ideal graphs of commutative rings, \textit{ Journal of Algebra and its Applications.}, \textbf{11}(06) (2012), 1250114.
%

\bibitem{hai2008finite}
Hai, ZHANG Qin and Ji, CAO Jian, Finite groups whose nontrivial normal subgroups have the same order, \textit{Journal of Mathematical Research and Exposition}, \textbf(28)(4) (2008), 807-812.
\end{thebibliography}

\end{document}

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