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\begin{document}
\title{Molecular description of Copper Oxide $CuO$
%, \thanks{This research
%is partially supported by COMSATS Attock and National University of Sciences and Technology, Islamabad, Pakistan}}
\author{Wei Gao$^1*$, Abdul Qudair Baig$^2$, Wasaq Khalid$^2$, , Mohammad Reza Farahani$^3$}}

\institute{$^1$School of Information Science and Technology, Yunnan Normal University,\\ Kunming 650500, China\\
$^2$Department of Mathematics, COMSATS Institute of Information Technology, \\
Attock Campus, Pakistan\\
$^3$Department of Applied Mathematics, Iran University of Science and Technology,\\ Narmak, 16844, Tehran, Iran\\
E-mail: gaowei@ynnu.edu.cn, \{aqbaig1@gmail.com, waqas.khalid38@gmail.com,
mrfarahani88\}@gmail.com}
\authorrunning{W. Gao, A.Q. Baig, W. Khalid, M.R. Farahani}}
\titlerunning{Molecular description of Copper Oxide $CuO$} \maketitle

\markboth{\small{W. Gao, A.Q. Baig, W. Khalid, M.R. Farahani}
{\small{Molecular description of Copper Oxide $CuO$}}

\begin{abstract}
Graph theory has many advancements in the field of mathematical chemistry. Recently, chemical graph theory is becoming very popular among researchers because of its wide applications in mathematical chemistry. The molecular topological descriptors are the numerical invariants of a molecular graph and are very useful to predict their bioactivity. A great variety of such indices are studied and used in theoretical chemistry, pharmaceutical researchers, in drugs and in different other fields. \\
In this article, we study the chemical graph of copper oxide and compute degree based topological indices mainly $ABC$, $GA$, $ABC_4$, $GA_5$, general Randi$\acute{c}$ index and Zagreb index for copper oxide $CuO$. Furthermore, we give an exact formulas of these indices which is helpful in studying the underlying topologies.
\end{abstract}
\par
\centerline{{\bf 2010 Mathematics Subject Classification:} 05C12, 05C90}
\vspace{.3cm} {\bf Keywords:} Molecular descriptor, Copper Oxide, Atom bond connectivity index, Geometric arithmetic index, General Randi$\acute{c}$ index, Zagreb Index, $ABC_4$, $GA_5$.

\section[Introduction]{Introduction}
There is a lot of chemical compounds either organic or inorganic possesses variety of commercial, industrial, pharmaceutical chemistry and laboratory importance. A relationship exists between chemical compounds and their molecular structures. Graph theory is a very powerful area of mathematics that has wide range of applications in many areas of science  such as chemistry, biology, computer science, electrical, electronics and other fields. The manipulation and examination of chemical structural information is made conceivable using molecular descriptors. Chemical graph theory is a branch of mathematical chemistry in which we apply tools of graph theory to model the chemical phenomenon mathematically. Furthermore, it relates with the nontrivial applications of graph theory for solving molecular problems. This theory contributes a prominent role in the field of chemical sciences. \\ \\
Cheminformatics is new subject which is a combination of chemistry, mathematics and information science. It examines (QSAR) and (QSPR) relationships that are utilized to foresee the biological activities and physiochemical properties of chemical compounds. A lot of research in the field of chemical graph theory has been done and researchers keep on researching in this field. Some references are given which is hopefully enough for the importance of this field \cite{imran,hayat,hayat1,hayat2,rajan,Baig,Farahani,S}. A chemical graph is a simple finite graph in which vertices denote the atoms and edges denote the chemical bonds in underlying chemical structure. A topological index is a numeric quantity associated with chemical constitution indicating for correlation of chemical structure with many physical, chemical properties and biological activities. \\ \\
The rapid increase in the diseases and environmental problems in our ecosystem give rise to increase in the physical and mental health problems of animals and human beings. Side by side, medicines manufacturing organizations and industries play their role to overcome diseases. The world health organization $WHO$ and certain institutions play their role to improve the environmental and health problems. A large number of medicines and drug products are produced each year. In order to determine the chemical properties of such medicines and drugs we focus on theoretical examination of topological indices. Smart polymer family is widely used in anticancer drugs and its some topological indices are computed in \cite{Gao}.
Let $G = (V,E)$ be a graph where $V$ be the vertex set and $E$ be the edge set of $G$. The degree $deg(v)$ of $v$ is the number of edges of $G$ incident with $v$. The distance $d(u,v)$ of a graph G is defined as the shortest length between $u$ and $v$. \\ \\
The concept of topological index came from work done by Wiener in 1947 while he was working on boiling point of paraffin \cite{Weiner}. Later on, the theory
of topological indices started. Let $G$ be a graph then Wiener index is defined as
\begin{equation*}
W(G)=\frac{1}{2}\sum_{u,v \in V(G)} d(u,v).
\end{equation*}
There are certain types of degree based topological indices. Some of them that are used in this article are defined below. \\
One of the first and oldest degree based index is introduced by $Milan$ \emph{Randi$\acute{c}$} \cite{randic} in 1975 and is defined below.
\begin{equation*}
R_{\frac{-1}{2}}(G)=\sum_{uv \in E(G)}\frac{1}{\sqrt{d_ud_v}}.
\end{equation*}
\noindent
In 1988, Bollob$\acute{a}$s $et$ $al.$ \cite{Bollo} and Amic $et$ $al.$ \cite{Amic} proposed the general Randi$\acute{c}$ index
independently. For more details about Randi$\acute{c}$ index, its properties and important results \cite{hu,li,Caporossi}. The general Randi$\acute{c}$ index is defined as
\begin{equation*}
R_{\alpha}(G)=\sum_{uv \in E(G)}(\sqrt{d_ud_v})^\alpha.
\end{equation*}
One of the important degree based topological index is the first Zagreb index. It was introduced in $1972$ by \cite{Gutm}. Later on, second Zagreb index is introduced by \cite{Gut}. Both first and second Zagreb index is formulated as
\begin{equation*}
M_1(G)=\sum_{uv \in E(G)} (d_u+d_v).
\end{equation*}
\begin{equation*}
M_2(G)=\sum_{uv \in E(G)} (d_ud_v).
\end{equation*}

\noindent
Among degree based topological indices, atom bond connectivity index of vital importance and introduced by Estrada $et$ $al.$ \cite{Estrada} and is defined as
\begin{equation*}
ABC(G)=\sum_{uv \in E(G)}\sqrt{\frac{d_u+d_v-2}{d_ud_v}}.
\end{equation*}
Where $d_u$ is the degree of vertex $u$. \\ \\
A well known topological index fourth version of atom bond connectivity index $ABC_4$ of a graph $G$ is introduced by Ghorbhani $et$ $al.$ \cite{ghorbani} and is defined as
\begin{equation*}
ABC_4(G)=\sum_{uv \in E(G)}\sqrt{\frac{S_u+S_v-2}{S_uS_v}}.
\end{equation*}
 Where $S_u = \sum_{uv \in E(G)}d_v$ and $S_v = \sum_{uv \in E(G)}d_u$. \\ \\
 The geometric arithmetic index $GA$ of a graph $G$ is introduced by Vuki$\check{c}$evi$\acute{c}$ $et$ $al.$ \cite{Vuki} and is defined as
 \begin{equation*}
 GA(G)=\sum_{uv \in E(G)}\frac{2\sqrt{d_ud_v}}{d_u+d_v}.
 \end{equation*}
 Another well known topological descriptor fifth version of geometric arithmetic index $GA_5$ of a graph $G$ is introduced by Graovoc $et$ $al$ \cite{Graovac} and is defined as
 \begin{equation*}
 GA_5(G)=\sum_{uv \in E(G)}\frac{2\sqrt{S_uS_v}}{S_u+S_v}.
 \end{equation*}
\begin{figure}[ht!]
\centering
\includegraphics[width=90mm]{ab}
\caption{(a) $Cu(OH)_2$ \,\,\,\,\ (b) $CuO$}
\label{overflow}
\end{figure}

\begin{figure}[ht!]
\centering
\includegraphics[width=60mm]{ass}
\caption{3D Copper oxide $CuO$.}
\label{overflow}
\end{figure}

\section{Copper Oxide/Cupric Oxide $CuO$}
In this section, some history about copper oxide is described and related applications to copper and copper oxide is provided in the form of references and the references therein. The copper oxide/cupric oxide is an inorganic chemical compound $CuO$. It is an essential mineral found in plants and animals. Copper has enormous applications in medical instruments, drugs, as a heat conductor and many others. Some applications of copper and cupric oxide is given in \cite{Cu1,Bio,Cu2,Baker}.\\ \\
In Fig 1(a), the copper hydroxide is depicted and when hydrogen atoms are depleted from $Cu(OH)_2$ then the resultant graph is depicted in Fig 1(b). The 3D graph of copper oxide $CuO$ is depicted in Fig 2. Copper oxide is used as the source of copper in mineral and vitamin supplements and is considered safe. Its use in medical devices, industrial and consumer products, is novel. The safety aspects of the use of copper oxide in products that come in contact with open and closed skin \cite{Bor}.


\section[Main results]{Main results}
In this section, we compute topological indices mainly the atom bond connectivity index, geometric arithmetic index, $ABC_4$, $GA_5$, general Randi$\acute{c}$ indices and Zagreb indices of copper oxide $CuO$. Furthermore, we compute a close formulas of these indices for $CuO$. A topological index can give us chemical information that is related to the chemical graph. It provides a numeric number which is helpful in $(QSAR/QSPR)$ studies. \\ \\
In this article, we consider the copper oxide molecular graph $CuO$ as depicted in figure 1(b). The construction of $CuO$ graph is such that the octagons are connected to each other in column wise and rows the connection between two octagons is achieved by making one $C_4$ between two octagons. For our convenience, we take $m$ and $n$ as the number of octagons in rows and columns respectively. The cardinality of vertices and edges in $CuO$ are $4mn+3n+m$ and $6mn+2n$ respectively. In $CuO$, the $2$ degree vertices are $mn+5n+2m$, the $3$ degree vertices are $2mn-2n$ and $4$ degree vertices are $mn-m$. \\ \\

\noindent
A close result formula of atom bond connectivity index for $CuO$ is computed in the following theorem.


 \begin{theorem}
Consider the graph of $G=CuO$ with $m,n>1$, then its atom bond connectivity index is equal to \\
$ABC(G)=m\Big\{2\sqrt{2}-\frac{2\sqrt{15}}{3}\Big\}+n\Big\{3\sqrt{2}-\frac{2\sqrt{15}}{3}\Big\}+mn\Big\{\sqrt{2}+\frac{2\sqrt{15}}{3}\Big\}-
2\sqrt{2}+\frac{2\sqrt{15}}{3}.$
\end{theorem}
\begin{proof}
We prove the above result by using Table 1 in the formula of atom bond connectivity index.
  \begin{equation*}
ABC(G)=\sum_{uv \in E(G)}\sqrt{\frac{d_u+d_v-2}{d_{u}d_{v}}}.
 \end{equation*}

\noindent
The following table shows the edge set partition of $G=CuO$.
\begin{table}[h!]
\centering
\begin{tabular}{|c|c|}
\hline
  $d_u,d_v$   &   Frequency   \\\hline
  $(2,2)$   & $4(n+1)$  \\\hline
  $(2,3)$   & $2mn+2(2m-n)-4$       \\\hline
  $(2,4)$   & $4(n-1)$        \\\hline
  $(3,4)$   & $4(mn-(m+n)+1)$    \\\hline
\end{tabular}
\caption{Edge partition of $CuO$ graph based on degrees of end vertices of each edge.}
\end{table}
\\
$ABC(G)$=$4(n+1)\sqrt{\frac{2+2-2}{2\times2}}+ (2mn+2(2m-n)-4)\sqrt{\frac{2+3-2}{2\times3}}+4(n-1)\sqrt{\frac{2+4-2}{2\times4}}+
4(mn-(m+n)+1)\sqrt{\frac{3+4-2}{3\times4}}$ \\ \\
After simplification and rearranging the terms, we get \\ \\
$ABC(G)=m\Big\{2\sqrt{2}-\frac{2\sqrt{15}}{3}\Big\}+n\Big\{3\sqrt{2}-\frac{2\sqrt{15}}{3}\Big\}+mn\Big\{\sqrt{2}+\frac{2\sqrt{15}}{3}\Big\}-
2\sqrt{2}+\frac{2\sqrt{15}}{3}.$ \qed
\end{proof}
A close result formula of geometric arithmetic index for $CuO$ is computed in the following theorem.

\begin{theorem}
Consider the graph of $G=CuO$ with $m,n>1$, then its geometric arithmetic index is equal to \\
$GA(G)=m\Big\{-\frac{16\sqrt{3}}{7}+\frac{8\sqrt{6}}{5}\Big\}+n\Big\{\frac{8\sqrt{2}}{3}-\frac{16\sqrt{3}}{7}-\frac{4\sqrt{6}}{5}+4\Big\}+
mn\Big\{\frac{16\sqrt{3}}{7}+\frac{4\sqrt{6}}{5}\Big\}-\frac{8\sqrt{2}}{3}+\frac{16\sqrt{3}}{7}-\frac{8\sqrt{6}}{5}+4.$
\end{theorem}
\begin{proof}
We prove the above result by using by using Table 1 in the formula of geometric arithmetic index.
\begin{equation*}
GA(G)=\sum_{uv \in E(G)} \frac{2\sqrt{d_ud_v}}{d_u+d_v}.
 \end{equation*}
 $GA(G)$=$4(n+1)\frac{2\sqrt{2\times2}}{2+2}+(2mn+2(2m-n)-4)\frac{2\sqrt{2\times3}}{2+3}+4(n-1)\frac{2\sqrt{2\times4}}{2+4}
+4(mn-(m+n)+1)\frac{2\sqrt{3\times4}}{3+4}$ \\ \\
After simplification and rearranging the terms, we get \\ \\
$GA(G)=m\Big\{-\frac{16\sqrt{3}}{7}+\frac{8\sqrt{6}}{5}\Big\}+n\Big\{\frac{8\sqrt{2}}{3}-\frac{16\sqrt{3}}{7}-\frac{4\sqrt{6}}{5}+4\Big\}+
mn\Big\{\frac{16\sqrt{3}}{7}+\frac{4\sqrt{6}}{5}\Big\}-\frac{8\sqrt{2}}{3}+\frac{16\sqrt{3}}{7}-\frac{8\sqrt{6}}{5}+4.$ \qed
\end{proof}
A close result formula of general Randi$\acute{c}$ index for $CuO$ is computed in the following theorem.

\begin{theorem}
Consider the graph of $G=CuO$ with $m,n>1$, then its general Randi$\acute{c}$ index is equal to \\
\begin{equation*}
R_{\alpha}(G) = \begin{cases}
60mn-12(2m+n)+8,                                         & $\text{if}$  \,\,\, \alpha=1, \\
\frac{1}{6}(4mn+2m+5n+1),                        &   $\text{if}$  \, \, \, \alpha=-1, \\
m\Big\{\frac{4\sqrt{3}-2\sqrt{6}}{3\sqrt{2}}\Big\}+n\Big\{\frac{2\sqrt{3}+2\sqrt{6}-2}{\sqrt{6}}\Big\}\\
+2mn\Big\{\frac{\sqrt{3}+\sqrt{6}}{3\sqrt{2}}\Big\}+\Big\{\frac{6\sqrt{2}+4\sqrt{3}+2\sqrt{6}-6}{3\sqrt{2}}\Big\}, & $\text{if}$ \, \, \,\alpha=\frac{-1}{2},\\
m(-8\sqrt{3}+4\sqrt{6})+n(8\sqrt{2}-8\sqrt{3}-2\sqrt{6}+8)\\+mn(8\sqrt{3}+2\sqrt{6})-8\sqrt{2}+8\sqrt{3}-4\sqrt{6}+8,        & $\text{if}$ \, \, \, \alpha=\frac{1}{2}.
\end{cases}
\end{equation*}

\end{theorem}
\begin{proof}
Let $G$ be a graph of $CuO$. To prove the above result, we use Table 1 in general Randi$\acute{c}$ index formula with $\alpha=1$.

\begin{equation*}
R_{1}(G)=\sum_{uv \in E(G)}(d_u \times d_v)
\end{equation*}
\noindent
$R_{1}(G)=4(n+1)(2\times2)+(2mn+2(2m-n)-4)(2\times3)+4(n-1)(2\times4)+4(mn-(m+n)+1)(3\times4)$ \\ \\
$R_{1}(G)=60mn-12(2m+n)+8$ \\ \\
For $\alpha=-1$, the formula of Randi$\acute{c}$ index takes the following form.
\begin{equation*}
R_{-1}(G)=\sum_{uv \in E(G)}\frac{1}{(d_u \times d_v)}
\end{equation*}
$R_{-1}(G)=4(n+1)\frac{1}{(2\times2)}+(2mn+(2m-n)-4)\frac{1}{(2\times3)}+4(n-1)\frac{1}{(2\times4)}+4(mn-(m+n)+1)\frac{1}{(3\times4)}$ \\ \\
$R_{-1}(G)=\frac{1}{6}(4mn+2m+5n+1)$ \\ \\
For $\alpha=\frac{1}{2}$, the formula of Randi$\acute{c}$ index takes the following form.
\begin{equation*}
R_{\frac{1}{2}}(G)=\sum_{uv \in E(G)}\sqrt{(d_u \times d_v)}
\end{equation*}
$R_{\frac{1}{2}}(G)=4(n+1)\sqrt{(2\times2)}+(2mn+2(2m-n)-4)\sqrt{(2\times3)}+4(n-1)\sqrt{(2\times4)}+4(mn-(m+n)+1)\sqrt{(3\times4)}$ \\ \\
$R_{\frac{1}{2}}(G)=m(-8\sqrt{3}+4\sqrt{6})+n(8\sqrt{2}-8\sqrt{3}-2\sqrt{6}+8)\\+mn(8\sqrt{3}+2\sqrt{6})-8\sqrt{2}+8\sqrt{3}-4\sqrt{6}+8$ \\ \\
For $\alpha=\frac{-1}{2}$, the formula of Randi$\acute{c}$ index takes the following form.
\begin{equation*}
R_{\frac{-1}{2}}(G)=\sum_{uv \in E(G)}\frac{1}{\sqrt{(d_u \times d_v)}}
\end{equation*}
$R_{\frac{-1}{2}}(G)=4(n+1)\frac{1}{\sqrt{(2\times2)}}+(2mn+2(2m-n)-4)\frac{1}{\sqrt{(2\times3)}}+4(n-1)\frac{1}{\sqrt{(2\times4)}}
+4(mn-(m+n)+1)\frac{1}{\sqrt{(3\times4)}}$ \\ \\
$R_{\frac{-1}{2}}(G)=m\Big\{\frac{4\sqrt{3}-2\sqrt{6}}{3\sqrt{2}}\Big\}+n\Big\{\frac{2\sqrt{3}+2\sqrt{6}-2}{\sqrt{6}}\Big\}
+2mn\Big\{\frac{\sqrt{3}+\sqrt{6}}{3\sqrt{2}}\Big\}+\Big\{\frac{6\sqrt{2}+4\sqrt{3}+2\sqrt{6}-6}{3\sqrt{2}}\Big\}.$ \qed
\end{proof}
\noindent
In the following theorem, we compute close results of first and second Zagreb indices for $CuO$.
\begin{theorem}
Consider the graph $G \cong CuO$, for $m,n>1$, then its first and second Zagreb index is equal to \\
\noindent
$M_1(G)=2(n-4m+19mn).$ \\ \\
$M_2(G)=60mn-12(2m+n)+8.$
\end{theorem}
\begin{proof}
Let $G$ be a graph of copper oxide $CuO$. The first Zagreb index can be calculated by using Table 1 in following equation. \\ \\
$M_1(G)=\sum_{uv \in E(G)} (d_u+d_v)$ \\ \\
\,\,\ $=4(n-1)(4)+(2mn+2(2m-n)-4)(5)+4(n-1)(6)+4(mn-(m+n)+1)(7)$ \\ \\

\noindent
$M_1(G)=2(n-4m+19mn).$ \\ \\
The second Zagreb index can be calculated by using Table 1. in following equation.\\ \\
 $M_2(G)=\sum_{uv \in E(G)} (d_u+d_v)$ \\ \\
 $=4(n-1)(4)+(2mn+2(2m-n)-4)(6)+4(n-1)(8)+4(mn-(m+n)+1)(12)$ \\ \\

$=60mn-12(2m+n)+8$.
\qed
\end{proof}
\noindent
A close result formula of fifth geometric arithmetic index for $CuO$ is computed in the following theorem.

\begin{theorem}
Consider the graph of $G=CuO$ with $m\geq3$ and $n>2$, then its fifth geometric arithmetic index is equal to \\
$GA_5(G)=m\Big\{6-\frac{\sqrt{15}}{2}-\frac{8\sqrt{30}}{11}\Big\}+n\Big\{4+\frac{8\sqrt{6}}{5}+\frac{\sqrt{15}}{2}-\frac{16\sqrt{30}}{11}\Big\}+
mn\Big\{\frac{\sqrt{15}}{2}+\frac{8\sqrt{30}}{11}\Big\}+\frac{16\sqrt{5}}{9}-\frac{8\sqrt{6}}{5}-\frac{\sqrt{15}}{2}+\frac{24\sqrt{30}}{11}-10.$
\end{theorem}
\begin{proof}
We prove the above result by using by using Table 2 in the formula of fifth geometric arithmetic index.
\begin{equation*}
GA_5(G)=\sum_{uv \in E(G)} \frac{2\sqrt{S_uS_v}}{S_u+S_v}.
 \end{equation*}
\\
\begin{table}[h!]
\centering
\begin{tabular}{|c|c|}
\hline
  $(S_u, S_v)$  &  Frequency        \\ \hline
   $(4,4)$      &     $4$          \\ \hline
   $(4,5)$      &    $4$           \\ \hline
   $(4,6)$      &    $4(n-1)$           \\ \hline
   $(5,6)$      &    $4$           \\ \hline
   $(6,6)$      &    $6m-10$           \\ \hline
   $(6,10)$     &    $2(mn-(m-n)-1)$           \\ \hline
   $(10,10)$    &    $4(n-1)$           \\ \hline
   $(10,12)$    &    $4(mn-(m+2n)+2)$           \\ \hline
\end{tabular}
\caption{Edge partition of $CuO$ graph based on degree sum of end vertices of each edge.}
\end{table}
\\
 $GA_5(G)$=$4\frac{2\sqrt{4\times4}}{4+4}+4\frac{2\sqrt{4\times5}}{4+5}+4(n-1)\frac{2\sqrt{4\times6}}{4+6}+4\frac{2\sqrt{5\times6}}{11}+
 (6m-10)\frac{2\sqrt{6\times6}}{6+6}+2(mn-(m-n)-1)\frac{2\sqrt{6\times10}}{6+10}+4(n-1)\frac{2\sqrt{10\times10}}{10+10}+
 4(mn-(m+2n)+2)\frac{2\sqrt{10\times12}}{10+12}$
\\ \\
After simplification and rearranging the terms, we get \\ \\
$GA_5(G)=m\Big\{6-\frac{\sqrt{15}}{2}-\frac{8\sqrt{30}}{11}\Big\}+n\Big\{4+\frac{8\sqrt{6}}{5}+\frac{\sqrt{15}}{2}-\frac{16\sqrt{30}}{11}\Big\}+
mn\Big\{\frac{\sqrt{15}}{2}+\frac{8\sqrt{30}}{11}\Big\}+\frac{16\sqrt{5}}{9}-\frac{8\sqrt{6}}{5}-\frac{\sqrt{15}}{2}+\frac{24\sqrt{30}}{11}-10.$ \qed
\end{proof}
\noindent
A close result formula of fourth atom bond connectivity index for $CuO$ is computed in the following theorem.



\begin{theorem}
Consider the graph of $G=CuO$ with $m\geq3$ and $n>2$, then its fourth atom bond connectivity index is equal to \\
$ABC_4(G)=m\Big\{\sqrt{10}-\frac{\sqrt{210}}{15}-\frac{4}{\sqrt{6}}\Big\}+n\Big\{\frac{6\sqrt{2}}{5}+\frac{4}{\sqrt{3}}+\frac{\sqrt{210}}{15}-
\frac{8}{\sqrt{6}}\Big\}+mn\Big\{\frac{\sqrt{210}}{15}+\frac{4}{\sqrt{6}}\Big\}-\frac{6\sqrt{2}}{5}-\frac{4}{\sqrt{3}}+\frac{14}{\sqrt{6}}
-\frac{5\sqrt{10}}{3}+\frac{2\sqrt{30}}{5}+\frac{2\sqrt{35}}{5}-\frac{\sqrt{210}}{15}.$
\end{theorem}
\begin{proof}
We prove the above result by using by using Table 2 in the formula of fourth atom bond connectivity index.
\begin{equation*}
ABC_4(G)=\sum_{uv \in E(G)} \sqrt{\frac{S_u+S_v-2}{S_uS_v}}.
 \end{equation*}
 $ABC_4(G)$=$4\sqrt{\frac{4+4-2}{4\times4}}+4\sqrt{\frac{4+5-2}{4\times5}}+4(n-1)\sqrt{\frac{4+6-2}{4\times6}}+4\sqrt{\frac{5+6-2}{5\times6}}+
 (6m-10)\sqrt{\frac{6+6-2}{6\times6}}+2(mn-(m-n)-1)\sqrt{\frac{6+10-2}{6\times10}}+4(n-1)\sqrt{\frac{10+10-2}{10\times10}}+
 4(mn-(m+2n)+2)\sqrt{\frac{10+12-2}{10\times12}}$
\\ \\
After simplification and rearranging the terms, we get \\ \\
$ABC_4(G)=m\Big\{\sqrt{10}-\frac{\sqrt{210}}{15}-\frac{4}{\sqrt{6}}\Big\}+n\Big\{\frac{6\sqrt{2}}{5}+\frac{4}{\sqrt{3}}+\frac{\sqrt{210}}{15}-
\frac{8}{\sqrt{6}}\Big\}+mn\Big\{\frac{\sqrt{210}}{15}+\frac{4}{\sqrt{6}}\Big\}-\frac{6\sqrt{2}}{5}-\frac{4}{\sqrt{3}}+\frac{14}{\sqrt{6}}
-\frac{5\sqrt{10}}{3}+\frac{2\sqrt{30}}{5}+\frac{2\sqrt{35}}{5}-\frac{\sqrt{210}}{15}.$ \qed
\end{proof}



\section[Conclusion]{Conclusion}
In this paper, we have computed some degree based topological indices
for chemical graph copper oxide graph $CuO$. An exact and close results
of atom bond connectivity index $ABC$, geometric
arithmetic index $GA$, general Randi$\acute{c}$ index, $GA_5$ and $ABC_4$, Zagreb indices for $CuO$.
These results are fruitful and helpful in understanding topological properties of cupric oxide.
In future, we are interested to sketch and design some new chemical graphs and examine their underlying topological properties.
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\end{document}
