# Computing Analysis of Connection-Based Indices and Coindices for Product of Molecular Networks

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## Abstract

**:**

## 1. Introduction

**Definition**

**1.**

- Either $[{a}_{1}={a}_{2}\in V({G}_{1})\wedge {b}_{1}{b}_{2}\in E({G}_{2})]$ or $[{b}_{1}={b}_{2}\in V({G}_{2})\wedge {a}_{1}{a}_{2}\in E({G}_{1})]$. For more detail, see Figure 1.

**Definition**

**2.**

**Definition**

**3.**

- Either $[{a}_{1}={a}_{2}\in V({G}_{1})\wedge {b}_{1}{b}_{2}\in E({G}_{2})]$ or $[{b}_{1},{b}_{2}\in V({G}_{2})\wedge {a}_{1}{a}_{2}\in E({G}_{1})]$. For more detail, see Figure 3.

## 2. Preliminaries

**Definition**

**4.**

**Definition**

**5.**

**Definition**

**6.**

**Definition**

**7.**

**Lemma**

**1**

**.**Let G be a connected network with n vertices and e edges. Subsequently, ${\tau}_{G}(a)+{d}_{G}(a)\le {\displaystyle \sum _{b\in {N}_{G}(a)}}{d}_{G}(b)$, where equality holds if and only if G is a $\{{C}_{3},{C}_{4}\}-$ free network.

**Lemma**

**2**

**.**Let G be a connected network with n vertices and e edges. Afterwards, $\sum _{b\in V(G)}}{d}_{G}(b)=2e$.

**Lemma**

**3**

**.**Let G be a connected network with n vertices and e edges. Subsequently, $\sum _{b\in V(G)}}{\tau}_{G}(b)\le {M}_{1}(G)-2e.$ where equality holds iff G is a $\{{C}_{3},{C}_{4}\}-$ free network.

## 3. A Few Molecular Networks

- Alkanes (hydrocarbon compounds) are organic compounds consisting of carbon atoms joined by single bounds. The simple and Lewis networks of alkanes are given in Figure 4. Moreover, methane ($C{H}_{4}$), ethane (${H}_{3}C-C{H}_{3}$), and propane (${H}_{3}C-C{H}_{2}-C{H}_{3}$) are examples of alkanes that are given in Figure 5. This alkane series continues and follows general formula as ${C}_{n}{H}_{2n+2}$.
- Cyclic compounds are molecules consisting of closed chain (ring) of at least three carbon atoms. If the closed chain has only carbon atoms, then it is an organic cyclic molecule that is called by homocyclic compound. If the closed chain has both carbon and non-carbon atoms, then it is an inorganic cyclic molecule that is called the heterocyclic compound. Moreover, Cycloalkanes (${C}_{n}{H}_{2n}$) are the isomers of alkenes consisting of exactly one cyclic compound joined by a single bond. Figure 6a,b presents the cyclic compounds (homocyclic and heterocyclic, respectively).

## 4. Main Results

**Theorem**

**1.**

**Proof.**

**a**). For $a\in V({G}_{1})$, $b\in V({G}_{2})$ and $(a,b)\in V({G}_{1}\times {G}_{2})$, we have, ${\tau}_{{G}_{1}\times {G}_{2}}(a,b)={\tau}_{{G}_{1}}(a)+{d}_{{G}_{1}}(a){d}_{{G}_{2}}(b)+{\tau}_{{G}_{2}}(b)$.

**b**).

**Theorem**

**2.**

**Proof.**

**a**). For $b\in V({G}_{1}\odot {G}_{2})$ either $b\in V({G}_{1})$ or $b\in V({G}_{2}^{i})$, where $1\le i\le {n}_{1}$.

**b**).

**Theorem**

**3.**

**Proof.**

**a**). For $a\in V({G}_{1})$, $b\in V({G}_{2})$ and $(a,b)\in V({G}_{1}\xb7{G}_{2})$, we have ${\tau}_{{G}_{1}[{G}_{2}]}(a,b)={n}_{2}{\tau}_{{G}_{1}}(a)+{d}_{\overline{{G}_{2}}}(b)={n}_{2}{\tau}_{{G}_{1}}(a)+({n}_{2}-1)-{d}_{{G}_{2}}(b)$.

**b**).

## 5. Applications, Comparisons and Conclusions

#### 5.1. Cartesian Product

**(1) Polynomial chains:**Let ${P}_{m}$ and ${P}_{n}$ be two particular path- alkanes, then the polynomial chains $({P}_{m}\times {P}_{n})$ are obtained by the Cartesian product of ${P}_{m}$ and ${P}_{n}$. For $m=6$ and $n=2$, see Figure 7.

- (
**a**) - $\overline{Z}{C}_{1}({P}_{m}\times {P}_{n})\le 2{m}^{2}n+2m{n}^{2}-4{m}^{2}-4{n}^{2}+8m+34n-32,$
- (
**b**) - $\overline{Z}{C}_{2}({P}_{m}\times {P}_{n})\le 2m{n}^{2}-6{n}^{2}+22mn-28m+60n-66.$

- $Z{C}_{1}^{*}$ of polynomial chains:
**(1)**If $m\ge 3$&$n=2$, $Z{C}_{1}^{*}({P}_{m}\times {P}_{n})\le 32mn-40m-42n+40$;**(2)**If $m\ge 3$&$n\ge 3$, $Z{C}_{1}^{*}({P}_{m}\times {P}_{n})\le 32mn-42m-42n+40$ - $Z{C}_{2}$ of polynomial chains:
**(1)**If $m\ge 3$&$n=2$, $Z{C}_{2}({P}_{m}\times {P}_{n})\le 120mn-192m-238n+350$;**(2)**If $m\ge 3$&$n=3$, $Z{C}_{2}({P}_{m}\times {P}_{n})\le 128mn-238m-246n+402$;**(3)**If $m\ge 3$&$n=4$, $Z{C}_{2}({P}_{m}\times {P}_{n})$ $\le 128mn-239m-246n+402$;**(4)**If $m\ge 5$&$n\ge 5$, $Z{C}_{2}({P}_{m}\times {P}_{n})\le 128mn-240m-246n+402$.

**(2) Carbon Nanotubes $(TU{C}_{4}(m,n))$:**Let ${P}_{m}$ and ${C}_{n}$ be a particular alkane and cycloalkane called by path and cycle, then carbon nanotubes $({P}_{m}\times {C}_{n})$ are obtained by the cartesian product of ${P}_{m}$ and ${C}_{n}$. For $m=4$ and $n=5$, see Figure 9.

- (
**a**) - $\overline{Z}{C}_{1}({P}_{m}\times {C}_{n})\le 2{m}^{2}n+2m{n}^{2}-4{n}^{2}+10mn-10n,$
- (
**b**) - $\overline{Z}{C}_{2}({P}_{m}\times {C}_{n})\le 2{m}^{2}n+2m{n}^{2}-6{n}^{2}+82mn-131n.$

- (
**1**) - $Z{C}_{1}^{*}({P}_{m}\times {C}_{n})\le 32mn-42n,$
- (
**2**) - $Z{C}_{2}({P}_{m}\times {C}_{n})\le 128mn-238n.$

#### 5.2. Corona Product

**(3) Alkane $({C}_{3}{H}_{8}):$**Let ${P}_{m}$ and ${N}_{n}$ be a particular alkane called by paths and a null graph, then the alkanes $({P}_{m}\odot {N}_{n})$ are obtained by the corona product of ${P}_{m}$ and ${N}_{n}$. The corona product only has a chemical sense when for arbitrary $m>0$, $n=2$, and $n=3$ provide equivalence chemical networks of alkenes and alkanes, respectively. Besides this sense, for $n>3$, see no chemical context of corona product. For $m=3$ and $n=3$, see Figure 11.

- (
**a**) - $\overline{Z}{C}_{1}({P}_{m}\odot {N}_{n})=mn+m-n-1,$
- (
**b**) - $\overline{Z}{C}_{2}({P}_{m}\odot {N}_{n})=m{n}^{2}-2{n}^{2}+mn+m-n-2.$

- (
**1**) - $Z{C}_{1}^{*}({P}_{m}\odot {N}_{n})=3m{n}^{2}-2{n}^{2}+7mn+4m-12n-10,$
- (
**2**) - $Z{C}_{2}({P}_{m}\odot {N}_{n})=2m{n}^{3}-2{n}^{3}+8m{n}^{2}-16{n}^{2}+10mn-26n.$

**(4) Cyclobutane (${C}_{4}{H}_{8}$):**Let ${C}_{m}$ and ${N}_{n}$ be a cycle and a null graph, then Cyclobutanes $({C}_{m}\odot {N}_{n})$ are obtained by the corona product of ${C}_{m}$ and ${N}_{n}$. The corona product has a chemical sense only when for arbitrary $m>0$, $n=1$ and $n=2$ provide equivalence chemical networks of cycloalkenes and cycloalkanes, respectively. Besides this sense, for $n>2$ see no chemical context (cyclic compounds) of corona product. For $m=4$ and $n=2$, see Figure 13.

- (
**a**) - $\overline{Z}{C}_{1}({C}_{m}\odot {N}_{n})\le 2mn+2m,$
- (
**b**) - $\overline{Z}{C}_{2}({C}_{m}\odot {N}_{n})\le 2m{n}^{2}+4mn+2m.$

- (
**1**) - $Z{C}_{1}^{*}({C}_{m}\odot {N}_{n})\le 3m{n}^{2}+7mn+4m,$
- (
**2**) - $Z{C}_{2}({C}_{m}\odot {N}_{n})\le 2m{n}^{3}+8m{n}^{2}+10mn+4m.$

#### 5.3. Lexicographic Product

**(5) Fence:**Let ${P}_{m}$ and ${P}_{n}$ be two particular path-alkanes, then the fence $({P}_{m}\xb7{P}_{n})$ are obtained by the lexicographic product of ${P}_{m}$ and ${P}_{n}$. For $m=6$ and $n=2$, see Figure 15.

- (
**a**) - $\overline{Z}{C}_{1}({P}_{m}\xb7{P}_{n})\le {m}^{2}{n}^{2}-3{m}^{2}n+m{n}^{2}+2{m}^{2}+4{n}^{2}+9mn-6m-6n+4,$
- (
**b**) - $\overline{Z}{C}_{2}({P}_{m}\xb7{P}_{n})\le \frac{{m}^{2}{n}^{3}}{2}-3{m}^{2}{n}^{2}+\frac{13}{2}{m}^{2}n-5{m}^{2}+\frac{13}{2}m{n}^{3}-3m{n}^{2}-\frac{23}{2}mn+15m-5{n}^{3}-12{n}^{2}+17n-10.$

- (
**1**) - $Z{C}_{1}^{*}({P}_{m}\xb7{P}_{n})=6m{n}^{3}-12{n}^{3}+4m{n}^{2}-6{n}^{2}-24mn+24m+20n-16,$
- (
**2**) - $Z{C}_{2}({P}_{m}\xb7{P}_{n})={n}^{5}+8m{n}^{4}-28{n}^{4}+5m{n}^{3}-6{n}^{3}-43m{n}^{2}+70{n}^{2}+71mn-46m-91n+34.$

**(6) Closed fence:**Let ${C}_{m}$ and ${P}_{n}$ be a cycle and a particular path-alkane, then closed fence $({C}_{m}\xb7{P}_{n})$ is obtained by the lexicographic product of ${C}_{m}$ and ${P}_{n}$. For $m=6$ and $n=2$, see Figure 17.

- (
**a**) - $\overline{Z}{C}_{1}({C}_{m}\xb7{P}_{n})\le {m}^{2}{n}^{2}-3{m}^{2}n+2{m}^{2}+3m{n}^{2}+9mn-6m,$
- (
**b**) - $\overline{Z}{C}_{2}({C}_{m}\xb7{P}_{n})\le \frac{{m}^{2}{n}^{3}}{2}-3{m}^{2}{n}^{2}+\frac{13}{2}{m}^{2}n-5{m}^{2}+\frac{21}{2}m{n}^{3}-9m{n}^{2}-\frac{15}{2}mn+15m.$

- (
**1**) - $Z{C}_{1}^{*}({C}_{m}\xb7{P}_{n})\le 4m{n}^{3}+4m{n}^{2}-24mn+24m,$
- (
**2**) - $Z{C}_{2}({C}_{m}\xb7{P}_{n})\le {n}^{5}+6m{n}^{4}-16{n}^{4}+7m{n}^{3}-8{n}^{3}-39m{n}^{2}+10{n}^{2}+67mn-46m-5n+2.$

- The behaviours of all the connection-based Zagreb indices and coindices for the molecular networks (polynomial chain, carbon nanotube, alkane, cycloalkane, fence, and closed fence) are symmetrise with some less or more values and the following orderings:(i) $Z{C}_{2}\ge \overline{Z}{C}_{2}\ge \overline{Z}{C}_{1}\ge Z{C}_{1}^{*}$ (for polynomial chain), (ii) $Z{C}_{2}\ge \overline{Z}{C}_{2}\ge Z{C}_{1}^{*}\ge \overline{Z}{C}_{1}$ (for carbon nanotubes, fence and closed fence) and (iii) $Z{C}_{2}\ge Z{C}_{1}^{*}\ge \overline{Z}{C}_{2}\ge \overline{Z}{C}_{1}$ (for alkane and cycloalkane).
- For increasing values of m and n in all of the molecular networks (polynomial chain, carbon nanotube, alkane, cycloalkane, fence, and closed fence), the second Zagreb connection index, and the first Zagreb connection coindex are responding rapidly, and steadily, respectively.
- In the certain intervals of the values of m and n, all the connection-based indices and coindices attain the maximum and minimum values. These values are also lifting up in the intervals on increasing values of m and n in such a way that the response of maximum values is more rapid than the minimum values. In addition, we analyse that second the Zagreb connection index has attained more upper layer than other TIs in all pf the molecular networks.
- In particular, Figure 19, Figure 20, Figure 21 and Figure 22 present that first Zagreb connection index, second Zagreb connection index, first Zagreb connection coindex, and second Zagreb connection coindex are dominant and auxiliary or incapable for the molecular networks from polynomial chain to closed fence, respectively. Moreover, we analyse that last molecular network i.e., closed fence has attain more upper layer than all other molecular networks for connection-based indices and coindices.

## Author Contributions

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**$(\mathbf{a})\phantom{\rule{0.166667em}{0ex}}{G}_{1}\cong {C}_{4}$, $(\mathbf{b})\phantom{\rule{0.166667em}{0ex}}{G}_{2}\cong {C}_{3}$ and $(\mathbf{c})\phantom{\rule{0.166667em}{0ex}}\mathrm{Cartesian}\mathrm{Product}\phantom{\rule{0.166667em}{0ex}}({C}_{4}\times {C}_{3}).$

**Figure 2.**$(\mathbf{d})\phantom{\rule{0.166667em}{0ex}}{G}_{1}\cong {C}_{6},(\mathbf{e})\phantom{\rule{0.166667em}{0ex}}{G}_{2}\cong {N}_{2}\phantom{\rule{0.166667em}{0ex}}\mathrm{and}\phantom{\rule{0.166667em}{0ex}}(\mathbf{f})\phantom{\rule{0.166667em}{0ex}}\mathrm{Cyclohexane}\phantom{\rule{0.166667em}{0ex}}({C}_{6}{H}_{12}={C}_{6}\odot {N}_{2}).$

**Figure 3.**$(\mathbf{g})\phantom{\rule{0.166667em}{0ex}}{G}_{1}\cong {C}_{4}$, $(\mathbf{h})\phantom{\rule{0.166667em}{0ex}}{G}_{2}\cong {P}_{3}$ and $(\mathbf{j})\phantom{\rule{0.166667em}{0ex}}\mathrm{Lexicographic}\phantom{\rule{0.166667em}{0ex}}\mathrm{Product}\phantom{\rule{0.166667em}{0ex}}({C}_{4}\xb7{P}_{3}).$

**Figure 4.**$(\mathbf{a})\phantom{\rule{0.166667em}{0ex}}{P}_{2},{P}_{3},{P}_{4}$ are simple networks of alkanes and $(\mathbf{b})\phantom{\rule{0.166667em}{0ex}}{P}_{2},{P}_{3},{P}_{4}$ are Lewis networks of alkanes.

**Figure 5.**Lewis network of $(\mathbf{a})$ Methane, $(\mathbf{b})$ Ethane and $(\mathbf{c})$ Propane.

**Figure 6.**$(\mathbf{a})$ The Lewis network of cyclopropane, cyclobutane, cyclopentane, and cyclohexane, $(\mathbf{b})$ The Lewis network of pyrol, thiophene, and pyridine.

**Figure 7.**$(\mathbf{a})$${H}_{1}\cong {P}_{6}$$(\mathbf{b})$${H}_{2}\cong {P}_{2}$ & $(\mathbf{c})$ Polynomial chain (${P}_{6}\times {P}_{2}$).

**Figure 8.**Polynomial chains of ${\theta}_{1}={P}_{m}\times {P}_{n}$ based on Table 1 with respect to indices and coindices.

**Figure 9.**$(\mathbf{a})$${H}_{1}\cong {P}_{4}$$(\mathbf{b})$${H}_{2}\cong {C}_{5}$ & $(\mathbf{c})$ Carbon nanotube $(TU{C}_{4}(m,n)\cong {P}_{4}\times {C}_{5})$.

**Figure 10.**Carbon nanotubes $(TU{C}_{4}(m,n))$ of ${\theta}_{2}={P}_{m}\times {C}_{n}$ based on Table 2 with respect to indices and coindices.

**Figure 11.**$(\mathbf{a})$${H}_{1}\cong {P}_{3}$$(\mathbf{b})$${H}_{2}\cong {N}_{3}$ & $(\mathbf{c})$ Alkane $({P}_{3}\odot {N}_{3}\sim {C}_{3}{H}_{8}$).

**Figure 12.**Alkanes of ${\theta}_{3}={P}_{m}\odot {N}_{n}$ based on Table 3 with respect to indices and coindices.

**Figure 13.**$(\mathbf{a})$${H}_{1}\cong {C}_{4}$$(\mathbf{b})$${H}_{2}\cong {N}_{2}$ & $(\mathbf{c})$ Cyclobutane $({C}_{4}\odot {N}_{2}\cong {C}_{4}{H}_{8}$).

**Figure 14.**Cyclobutanes of ${\theta}_{4}={C}_{m}\odot {N}_{n}$ based on Table 4 with respect to indices and coindices.

**Figure 15.**$(\mathbf{a})$${H}_{1}\cong {P}_{6}$$(\mathbf{b})$${H}_{2}\cong {P}_{2}$ & $(\mathbf{c})$ Fence $({P}_{6}\xb7{P}_{2}$).

**Figure 16.**Fence of ${\theta}_{5}={P}_{m}\xb7{P}_{n}$ based on Table 5 with respect to indices and coindices.

**Figure 17.**$(\mathbf{a})$${H}_{1}\cong {C}_{6}$$(\mathbf{b})$${H}_{2}\cong {P}_{2}$ & $(\mathbf{c})$ Closed fence $S({C}_{6}\xb7{P}_{2}$).

**Figure 18.**Closed fence of ${\theta}_{6}={C}_{m}\xb7{P}_{n}$ based on Table 6 with respect to indices and coindices.

(m,n) | ${\mathit{ZC}}_{1}^{*}({\mathit{\theta}}_{1})$ | ${\mathit{ZC}}_{2}({\mathit{\theta}}_{1})$ | $\overline{\mathit{Z}}{\mathit{C}}_{1}({\mathit{\theta}}_{1})$ | $\overline{\mathit{Z}}{\mathit{C}}_{2}({\mathit{\theta}}_{1})$ |
---|---|---|---|---|

(3,2) | 28 | 18 | 68 | 102 |

(3,3) | 76 | 102 | 130 | 228 |

(3,4) | 130 | 237 | 196 | 354 |

(3,5) | 184 | 372 | 266 | 480 |

(4,2) | 52 | 66 | 84 | 126 |

(4,3) | 130 | 248 | 170 | 284 |

(4,4) | 216 | 510 | 264 | 446 |

(4,5) | 302 | 772 | 366 | 612 |

(5,2) | 76 | 114 | 100 | 150 |

(5,3) | 184 | 394 | 214 | 340 |

(5,4) | 302 | 783 | 340 | 538 |

(5,5) | 420 | 1172 | 478 | 744 |

(6,2) | 100 | 162 | 116 | 174 |

(6,3) | 238 | 540 | 262 | 396 |

(6,4) | 388 | 1056 | 424 | 630 |

(6,5) | 538 | 1572 | 602 | 876 |

(m,n) | ${\mathit{ZC}}_{1}^{*}({\mathit{\theta}}_{2})$ | ${\mathit{ZC}}_{2}({\mathit{\theta}}_{2})$ | $\overline{\mathit{Z}}{\mathit{C}}_{1}({\mathit{\theta}}_{2})$ | $\overline{\mathit{Z}}{\mathit{C}}_{2}({\mathit{\theta}}_{2})$ |
---|---|---|---|---|

(3,2) | 108 | 292 | 84 | 266 |

(3,3) | 162 | 438 | 132 | 399 |

(3,4) | 216 | 584 | 184 | 532 |

(3,5) | 270 | 730 | 240 | 665 |

(4,2) | 172 | 548 | 140 | 466 |

(4,3) | 258 | 822 | 222 | 705 |

(4,4) | 344 | 1096 | 312 | 948 |

(4,5) | 430 | 1370 | 410 | 1195 |

(5,2) | 236 | 804 | 204 | 674 |

(5,3) | 354 | 1206 | 324 | 1023 |

(5,4) | 472 | 1608 | 456 | 1380 |

(5,5) | 590 | 2010 | 600 | 1745 |

(6,2) | 300 | 1060 | 276 | 890 |

(6,3) | 450 | 1590 | 438 | 1353 |

(6,4) | 600 | 2120 | 616 | 1828 |

(6,5) | 750 | 2650 | 810 | 2315 |

(m,n) | ${\mathit{ZC}}_{1}^{*}({\mathit{\theta}}_{3})$ | ${\mathit{ZC}}_{2}({\mathit{\theta}}_{3})$ | $\overline{\mathit{Z}}{\mathit{C}}_{1}({\mathit{\theta}}_{3})$ | $\overline{\mathit{Z}}{\mathit{C}}_{2}({\mathit{\theta}}_{3})$ |
---|---|---|---|---|

(3,2) | 48 | 72 | 6 | 9 |

(3,3) | 92 | 192 | 8 | 16 |

(3,4) | 150 | 400 | 10 | 25 |

(3,5) | 222 | 720 | 12 | 36 |

(4,2) | 78 | 140 | 9 | 16 |

(4,3) | 144 | 348 | 12 | 29 |

(4,4) | 230 | 696 | 15 | 46 |

(4,5) | 336 | 1220 | 18 | 67 |

(5,2) | 108 | 208 | 12 | 23 |

(5,3) | 196 | 504 | 16 | 42 |

(5,4) | 310 | 992 | 20 | 67 |

(5,5) | 450 | 1720 | 24 | 98 |

(6,2) | 138 | 276 | 15 | 30 |

(6,3) | 248 | 660 | 20 | 55 |

(6,4) | 390 | 1288 | 25 | 88 |

(6,5) | 564 | 2220 | 30 | 129 |

(m,n) | ${\mathit{ZC}}_{1}^{*}({\mathit{\theta}}_{4})$ | ${\mathit{ZC}}_{2}({\mathit{\theta}}_{4})$ | $\overline{\mathit{Z}}{\mathit{C}}_{1}({\mathit{\theta}}_{4})$ | $\overline{\mathit{Z}}{\mathit{C}}_{2}({\mathit{\theta}}_{4})$ |
---|---|---|---|---|

(3,2) | 90 | 216 | 18 | 54 |

(3,3) | 156 | 480 | 24 | 96 |

(3,4) | 240 | 900 | 30 | 150 |

(3,5) | 342 | 1512 | 36 | 216 |

(4,2) | 120 | 288 | 24 | 72 |

(4,3) | 208 | 640 | 32 | 128 |

(4,4) | 320 | 1200 | 40 | 200 |

(4,5) | 456 | 2016 | 48 | 288 |

(5,2) | 150 | 360 | 30 | 90 |

(5,3) | 260 | 800 | 40 | 160 |

(5,4) | 400 | 1500 | 50 | 250 |

(5,5) | 570 | 2520 | 60 | 360 |

(6,2) | 180 | 432 | 36 | 100 |

(6,3) | 312 | 960 | 48 | 174 |

(6,4) | 480 | 1800 | 60 | 268 |

(6,5) | 684 | 3024 | 72 | 382 |

(m,n) | ${\mathit{ZC}}_{1}^{*}({\mathit{\theta}}_{5})$ | ${\mathit{ZC}}_{2}({\mathit{\theta}}_{5})$ | $\overline{\mathit{Z}}{\mathit{C}}_{1}({\mathit{\theta}}_{5})$ | $\overline{\mathit{Z}}{\mathit{C}}_{2}({\mathit{\theta}}_{5})$ |
---|---|---|---|---|

(3,2) | 24 | -56 | 56 | 32 |

(3,3) | 116 | -107 | 130 | 194 |

(3,4) | 328 | 16 | 236 | 602 |

(3,5) | 696 | 781 | 374 | 1370 |

(4,2) | 64 | 36 | 72 | 64 |

(4,3) | 266 | 456 | 174 | 330 |

(4,4) | 704 | 1934 | 324 | 974 |

(4,5) | 1450 | 5640 | 522 | 2170 |

(5,2) | 104 | 128 | 88 | 96 |

(5,3) | 416 | 1019 | 222 | 468 |

(5,4) | 1080 | 3852 | 424 | 1356 |

(5,5) | 2204 | 10499 | 694 | 3000 |

(6,2) | 144 | 220 | 104 | 128 |

(6,3) | 566 | 1582 | 274 | 608 |

(6,4) | 1456 | 5770 | 536 | 1748 |

(6,5) | 2958 | 15358 | 890 | 3860 |

(m,n) | ${\mathit{ZC}}_{1}^{*}({\mathit{\theta}}_{6})$ | ${\mathit{ZC}}_{2}({\mathit{\theta}}_{6})$ | $\overline{\mathit{Z}}{\mathit{C}}_{1}({\mathit{\theta}}_{6})$ | $\overline{\mathit{Z}}{\mathit{C}}_{2}({\mathit{\theta}}_{6})$ |
---|---|---|---|---|

(3,2) | 72 | -4 | 72 | 144 |

(3,3) | 288 | 245 | 162 | 594 |

(3,4) | 744 | 1304 | 288 | 1584 |

(3,5) | 1512 | 4169 | 450 | 3330 |

(4,2) | 96 | 80 | 96 | 192 |

(4,3) | 384 | 724 | 224 | 796 |

(4,4) | 992 | 2886 | 408 | 2132 |

(4,5) | 2016 | 8108 | 648 | 4500 |

(5,2) | 120 | 164 | 120 | 240 |

(5,3) | 480 | 1203 | 290 | 1000 |

(5,4) | 1240 | 4468 | 540 | 2690 |

(5,5) | 2520 | 12047 | 870 | 5700 |

(6,2) | 144 | 248 | 144 | 288 |

(6,3) | 576 | 1682 | 360 | 1206 |

(6,4) | 1488 | 6050 | 684 | 3258 |

(6,5) | 3024 | 15986 | 1116 | 6930 |

© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Ali, U.; Javaid, M.; Alanazi, A.M.
Computing Analysis of Connection-Based Indices and Coindices for Product of Molecular Networks. *Symmetry* **2020**, *12*, 1320.
https://doi.org/10.3390/sym12081320

**AMA Style**

Ali U, Javaid M, Alanazi AM.
Computing Analysis of Connection-Based Indices and Coindices for Product of Molecular Networks. *Symmetry*. 2020; 12(8):1320.
https://doi.org/10.3390/sym12081320

**Chicago/Turabian Style**

Ali, Usman, Muhammad Javaid, and Abdulaziz Mohammed Alanazi.
2020. "Computing Analysis of Connection-Based Indices and Coindices for Product of Molecular Networks" *Symmetry* 12, no. 8: 1320.
https://doi.org/10.3390/sym12081320