**5.** *TUC***4***C***8[***p***,** *q***] Nanotube and Nanotorus**

We will use the notations and notions of Diudea and Graovac, and the 2D lattice of *TUC*4*C*8[*p*, *q*] Nanotorus is denoted by *KTUC*[*p*, *q*] (see Figure 5) and the *TUC*4*C*8[*p*, *q*] Nanotube is denoted by *GTUC*[*p*, *q*] (see Figure 6). A *TUC*4*C*8[*p*, *q*] Nanotube is constructed in such a way that the total cardinality of octagons in each row equalsp *p* and the total cardinality of octagons in each column equals *q*. An example is presented in Figure 6. In *TUC*4*C*8[*p*, *q*] Nanotube, the total cardinality of octagons and squares are the same as those in each row, and in *TUC*4*C*8[*p*, *q*] Nanotorus the total cardinality of octagons and squares are the same as those in each row and column. In 2D lattice of *TUC*4*C*8[*p*, *q*] Nanotorus, the total cardinality of squares in rows and columns are (*p* + 1) and (*q* + <sup>1</sup>), respectively (cf. [30,31]).

The cardinalities of the vertex and edge set of *KTUC*[*p*, *q*] and *GTUC*[*p*, *q*] are presented in the following Table 1.

**Table 1.** Order and size of Nanotorus *KTUC*[*p*, *q*] and Nanotube *GTUC*[*p*, *q*].


**Figure 5.** 2D-lattice of *TUC*4*C*8(*R*)[*p*, *q*] Nanotorus with *p* = 5 and *q* = 3.

### *5.1. Methodology of KTUC*[*p*, *q*],(*p*, *q* ≥ 1) *Nanotorus Formulas*

For the Nanotorus *KTUC*[*p*, *q*],(*p*, *q* ≥ <sup>1</sup>), we have that the number of vertices in *KTUC*[*p*, *q*] is <sup>4</sup>*p*(*p* + <sup>1</sup>)(*q* + 1) and the number of edges is 6*pq* + <sup>5</sup>(*p* + *q*) + 4. The edge set can be partitioned into the following six disjoint sets:

*<sup>E</sup>*1*KTUC*[*p*, *q*] = *sr* ∈ *<sup>E</sup>KTUC*[*p*, *q*] | *dgr*(*s*) = 5, *dgr*(*r*) = 5 *<sup>E</sup>*2*KTUC*[*p*, *q*] = *sr* ∈ *<sup>E</sup>KTUC*[*p*, *q*] | *dgr*(*s*) = 5, *dgr*(*r*) = 8 *<sup>E</sup>*3*KTUC*[*p*, *q*] = *sr* ∈ *<sup>E</sup>KTUC*[*p*, *q*] | *dgr*(*s*) = 6, *dgr*(*r*) = 8 *<sup>E</sup>*4*KTUC*[*p*, *q*] = *sr* ∈ *<sup>E</sup>KTUC*[*p*, *q*] | *dgr*(*s*) = 8, *dgr*(*r*) = 8 *<sup>E</sup>*5*KTUC*[*p*, *q*] = *sr* ∈ *<sup>E</sup>KTUC*[*p*, *q*] | *dgr*(*s*) = 8, *dgr*(*r*) = 9 *<sup>E</sup>*6*KTUC*[*p*, *q*] = *sr* ∈ *<sup>E</sup>KTUC*[*p*, *q*] | *dgr*(*s*) = 9, *dgr*(*r*) = 9

We can obtain that |*<sup>E</sup>*1*KTUC*[*p*, *q*]| = 4, |*<sup>E</sup>*2*KTUC*[*p*, *q*]| = 8, |*<sup>E</sup>*3*KTUC*[*p*, *q*]| = <sup>4</sup>(*p* + *q* − <sup>2</sup>), |*<sup>E</sup>*4*KTUC*[*p*, *q*]| = <sup>2</sup>(*p* + *q* + <sup>2</sup>), |*<sup>E</sup>*5*KTUC*[*p*, *q*]| = <sup>4</sup>(*p* + *q* − 2) and |*<sup>E</sup>*6*KTUC*[*p*, *q*]| = 6*pq* − 5*p* − 5*q* + 4, and the representatives of these partitioned edge set are demonstrated in Figure 5, in which the edge set with color red, green, blue, yellow, brown and black are *<sup>E</sup>*1*KTUC*[*p*, *<sup>q</sup>*], *<sup>E</sup>*2*KTUC*[*p*, *<sup>q</sup>*], *<sup>E</sup>*3*KTUC*[*p*, *<sup>q</sup>*], *<sup>E</sup>*4*KTUC*[*p*, *<sup>q</sup>*], *<sup>E</sup>*5*KTUC*[*p*, *q*] and *<sup>E</sup>*6*KTUC*[*p*, *q*] respectively.

*5.2. Main Results for KTUC*[*p*, *q*],(*p*, *q* ≥ 1) *Nanotorus*

### • **Hyper Zagreb index of** *KTUC***[***p***,** *q***], (***p***,** *q ≥* **1) Nanotorus**

Let *G* = *KTUC*[*p*, *q*]. Now using Equation (1), we have

$$\begin{split} HM(G) &= \sum\_{sr \in \mathbb{E}(G)} \left[ dgr(s) + dgr(r) \right]^2 \\ HM(K) &= \sum\_{sr \in \mathbb{E}\_1} \left[ dgr(s) + dgr(r) \right]^2 + \sum\_{sr \in \mathbb{E}\_2} \left[ dgr(s) + dgr(r) \right]^2 + \sum\_{sr \in \mathbb{E}\_3} \left[ dgr(s) + dgr(r) \right]^2 \\ &+ \sum\_{sr \in \mathbb{E}\_4} \left[ dgr(s) + dgr(r) \right]^2 + \sum\_{sr \in \mathbb{E}\_5} \left[ dgr(s) + dgr(r) \right]^2 + \sum\_{sr \in \mathbb{E}\_6} \left[ dgr(s) + dgr(r) \right]^2 \\ &= 10^2 |E\_1| + 13^2 |E\_2| + 14^2 |E\_3| + 16^2 |E\_4| + 17^2 |E\_5| + 18^2 |E\_6| \\ &= 100(4) + 169(8) + 196(4p + 4q - 8) + 256(2p + 2q + 4) \\ &+ 289(4p + 4q - 8) + 324(6pq - 5p - 5q + 4) \\ &= 1944pq + 832(p + q) + 192 \end{split}$$

### • **Multiple Zagreb indices of** *KTUC***[***p***,** *q***], (***p***,** *q ≥* **1) Nanotorus**

Let *G* = *KTUC*[*p*, *q*]. Now using Equations (2) and (3) we have

$$\begin{array}{lcl} PM1(G) &=& \prod\_{sr \in E(G)} \left[ dgr(s) + dgr(r) \right] \\ PM1(K) &=& \prod\_{sr \in E\_1} \left[ dgr(s) + dgr(r) \right] \times \prod\_{sr \in E\_2} \left[ dgr(s) + dgr(r) \right] \times \prod\_{sr \in E\_3} \left[ dgr(s) + dgr(r) \right] \\ & \times \prod\_{sr \in E\_4} \left[ dgr(s) + dgr(r) \right] \times \prod\_{sr \in E\_5} \left[ dgr(s) + dgr(r) \right] \times \prod\_{sr \in E\_6} \left[ dgr(s) + dgr(r) \right] \\ &=& 10^{|E\_1|} \times 13^{|E\_2|} \times 14^{|E\_3|} \times 16^{|E\_4|} \times 17^{|E\_5|} \times 18^{|E\_6|} \\ &=& 10^4 \times 13^8 \times 14^{(4p + 4q - 8)} \times 16^{(2p + 2q + 4)} \times 17^{(4p + 4q - 8)} \times 18^{(6p - 5p - 5q + 4)} \\ \end{array}$$

$$\begin{split} PM\_{2}(\mathbb{G}) &= \prod\_{\mathcal{S}\in\mathcal{E}(G)} [\mathcal{d}gr(\boldsymbol{s}) \times \mathcal{d}gr(\boldsymbol{r})] \\ PM\_{2}(\mathbb{K}) &= \prod\_{\mathcal{S}\in\mathcal{E}\_{1}} \left[ \mathcal{d}gr(\boldsymbol{s}) \times \mathcal{d}gr(\boldsymbol{r}) \right] \times \prod\_{\mathcal{S}\in\mathcal{E}\_{2}} \left[ \mathcal{d}gr(\boldsymbol{s}) \times \mathcal{d}gr(\boldsymbol{r}) \right] \times \prod\_{\mathcal{S}\in\mathcal{E}\_{3}} \left[ \mathcal{d}gr(\boldsymbol{s}) \times \mathcal{d}gr(\boldsymbol{r}) \right] \\ &\times \prod\_{\mathcal{S}\in\mathcal{E}\_{4}} \left[ \mathcal{d}gr(\boldsymbol{s}) \times \mathcal{d}gr(\boldsymbol{r}) \right] \times \prod\_{\mathcal{S}\in\mathcal{E}\_{5}} \left[ \mathcal{d}gr(\boldsymbol{s}) \times \mathcal{d}gr(\boldsymbol{r}) \right] \times \prod\_{\mathcal{S}\in\mathcal{E}\_{6}} \left[ \mathcal{d}gr(\boldsymbol{s}) \times \mathcal{d}gr(\boldsymbol{r}) \right] \\ &= 25^{|\mathcal{E}\_{1}|} \times 40^{|\mathcal{E}\_{2}|} \times 48^{|\mathcal{E}\_{3}|} \times 64^{|\mathcal{E}\_{4}|} \times 72^{|\mathcal{E}\_{5}|} \times 81^{|\mathcal{E}\_{6}|} \\ &= 25^{4} \times 40^{8} \times 48^{(4p+4q-8)} \times 64^{(2p+2q+4)} \times 72^{(4p+4q-8)} \times 81^{(6p-5p-5q+4)} \end{split}$$

### • **Zagreb polynomials of** *KTUC***[***p***,** *q***], (***p***,** *q ≥* **1) Nanotorus**

Let *G* = *KTUC*[*p*, *q*]. Now using Equations (4) and (5) we have

$$\begin{array}{rcl} M\_1(G, \boldsymbol{x}) &=& \sum\_{\boldsymbol{s}\boldsymbol{r}\in E(G)} \boldsymbol{x}^{[\boldsymbol{d}\boldsymbol{g}\boldsymbol{r}(\boldsymbol{s}) + \boldsymbol{d}\boldsymbol{g}\boldsymbol{r}(\boldsymbol{r})]} \\\\ M\_1(K, \boldsymbol{\chi}) &=& \sum\_{\boldsymbol{s}\boldsymbol{r}\in E\_1} \boldsymbol{x}^{[\boldsymbol{d}\boldsymbol{g}\boldsymbol{r}(\boldsymbol{s}) + \boldsymbol{d}\boldsymbol{g}\boldsymbol{r}(\boldsymbol{r})]} + \sum\_{\boldsymbol{s}\boldsymbol{r}\in E\_2} \boldsymbol{x}^{[\boldsymbol{d}\boldsymbol{g}\boldsymbol{r}(\boldsymbol{s}) + \boldsymbol{d}\boldsymbol{g}\boldsymbol{r}(\boldsymbol{r})]} + \sum\_{\boldsymbol{s}\boldsymbol{r}\in E\_3} \boldsymbol{x}^{[\boldsymbol{d}\boldsymbol{g}\boldsymbol{r}(\boldsymbol{s}) + \boldsymbol{d}\boldsymbol{g}\boldsymbol{r}(\boldsymbol{r})]} \\\\ &+& \sum\_{\boldsymbol{s}\boldsymbol{r}\in E\_4} \boldsymbol{x}^{[\boldsymbol{d}\boldsymbol{g}\boldsymbol{r}(\boldsymbol{s}) + \boldsymbol{d}\boldsymbol{g}\boldsymbol{r}(\boldsymbol{r})]} + \sum\_{\boldsymbol{s}\boldsymbol{r}\in E\_5} \boldsymbol{x}^{[\boldsymbol{d}\boldsymbol{g}\boldsymbol{r}(\boldsymbol{s}) + \boldsymbol{d}\boldsymbol{g}\boldsymbol{r}(\boldsymbol{r})]} + \sum\_{\boldsymbol{s}\boldsymbol{r}\in E\_6} \boldsymbol{x}^{[\boldsymbol{d}\boldsymbol{g}\boldsymbol{r}(\boldsymbol{s}) + \boldsymbol{d}\boldsymbol{g}\boldsymbol{r}(\boldsymbol{r})]} \end{array}$$

$$\begin{split} M\_{1}(\mathbf{X},\mathbf{x}) &= \sum\_{s\tau\in E\_{1}} \mathbf{x}^{10} + \sum\_{s\tau\in E\_{2}} \mathbf{x}^{13} + \sum\_{s\tau\in E\_{3}} \mathbf{x}^{14} + \sum\_{s\tau\in E\_{4}} \mathbf{x}^{16} + \sum\_{s\tau\in E\_{5}} \mathbf{x}^{17} + \sum\_{s\tau\in E\_{6}} \mathbf{x}^{18} \\ &= \|E\_{1}|\mathbf{x}^{10} + |E\_{2}|\mathbf{x}^{13} + |E\_{3}|\mathbf{x}^{14} + |E\_{4}|\mathbf{x}^{16} + |E\_{5}|\mathbf{x}^{17} + |E\_{6}|\mathbf{x}^{18} \\ &= 4\mathbf{x}^{10} + 8\mathbf{x}^{13} + (4p + 4q - 8)\mathbf{x}^{14} + (2p + 2q + 4)\mathbf{x}^{16} \\ &\quad + (4p + 4q - 8)\mathbf{x}^{17} + (6pq - 5p - 5q + 4)\mathbf{x}^{18} \end{split}$$

*<sup>M</sup>*2(*<sup>G</sup>*, *x*) = ∑ *sr*∈*<sup>E</sup>*(*G*) *x*[*dgr*(*s*)×*dgr*(*r*)] *<sup>M</sup>*2(*<sup>K</sup>*, *x*) = ∑ *sr*∈*E*1 *x*[*dgr*(*s*)×*dgr*(*r*)] + ∑ *sr*∈*E*2 *x*[*dgr*(*s*)×*dgr*(*r*)] + ∑ *sr*∈*E*3 *x*[*dgr*(*s*)×*dgr*(*r*)] + ∑ *sr*∈*E*4 *x*[*dgr*(*s*)×*dgr*(*r*)] + ∑ *sr*∈*E*5 *x*[*dgr*(*s*)×*dgr*(*r*)] + ∑ *sr*∈*E*6 *x*[*dgr*(*s*)×*dgr*(*r*)] = ∑ *sr*∈*E*1 *x*<sup>25</sup> + ∑ *sr*∈*E*2 *x*<sup>40</sup> + ∑ *sr*∈*E*3 *x*<sup>48</sup> + ∑ *sr*∈*E*4 *x*<sup>64</sup> + ∑ *sr*∈*E*5 *x*<sup>72</sup> + ∑ *sr*∈*E*6 *x*<sup>81</sup> = |*<sup>E</sup>*1|*x*<sup>25</sup> + |*<sup>E</sup>*2|*x*<sup>40</sup> + |*<sup>E</sup>*3|*x*<sup>48</sup> + |*<sup>E</sup>*4|*x*<sup>64</sup> + |*<sup>E</sup>*5|*x*<sup>72</sup> + |*<sup>E</sup>*6|*x*<sup>81</sup> = 4*x*<sup>25</sup> + 8*x*<sup>40</sup> + (<sup>4</sup>*p* + 4*q* − <sup>8</sup>)*x*<sup>48</sup> + (<sup>2</sup>*p* + 2*q* + <sup>4</sup>)*x*<sup>64</sup> + (<sup>4</sup>*p* + 4*q* − <sup>8</sup>)*x*<sup>72</sup> + (<sup>6</sup>*pq* − 5*p* − 5*q* + <sup>4</sup>)*x*<sup>81</sup>

### *5.3. Methodology and Results of GTUC*[*p*, *q*],(*p*, *q* ≥ 1) *Nanotube Formulas*

For the Nanotube *GTUC*[*p*, *q*],(*p*, *q* ≥ <sup>1</sup>), we know that the number of vertices in *GTUC*[*p*, *q*] are <sup>4</sup>*p*(*q* + 1) and the number of edges are 6*pq* + 5*p*. The edge set can be partitioned into the following four disjoint sets:

*<sup>E</sup>*1*GTUC*[*p*, *q*] = *sr* ∈ *<sup>E</sup>GTUC*[*p*, *q*] | *dgr*(*s*) = 6, *dgr*(*r*) = 8 *<sup>E</sup>*2*GTUC*[*p*, *q*] = *sr* ∈ *<sup>E</sup>GTUC*[*p*, *q*] | *dgr*(*s*) = 8, *dgr*(*r*) = 8 *<sup>E</sup>*3*GTUC*[*p*, *q*] = *sr* ∈ *<sup>E</sup>GTUC*[*p*, *q*] | *dgr*(*s*) = 8, *dgr*(*r*) = 9 *<sup>E</sup>*4*GTUC*[*p*, *q*] = *sr* ∈ *<sup>E</sup>GTUC*[*p*, *q*] | *dgr*(*s*) = 9, *dgr*(*r*) = 9 

The cardinality of edges in *<sup>E</sup>*1*GTUC*[*p*, *q*]are 4*p*, in *<sup>E</sup>*2*GTUC*[*p*, *q*]are 2*p*, in *<sup>E</sup>*3*GTUC*[*p*, *q*]are 4*p* and in *<sup>E</sup>*4*GTUC*[*p*, *q*] are 6*pq* − 5*p*. The representatives of these edge set partitions are shown in Figure 6 in which red, green, blue and black edges are the edges belong to *<sup>E</sup>*1*GTUC*[*p*, *<sup>q</sup>*], *<sup>E</sup>*2*GTUC*[*p*, *<sup>q</sup>*], *<sup>E</sup>*3*GTUC*[*p*, *q*] and *<sup>E</sup>*4*GTUC*[*p*, *q*] respectively. Now using Equations (1)–(5), we have

**Figure 6.** Nanotube *TUC*4*C*8(*R*)[*p*, *q*] Nanotube with *p* = 5 and *q* = 4.

### • **Hyper Zagreb index of** *GTUC***[***p***,** *q***], (***p***,** *q ≥* **1) Nanotube**

Let *G* = *GTUC*[*p*, *q*]. Now using Equation (1), we have

$$\begin{aligned} HM(G) &= \sum\_{sr \in E(G)} \left[ dgr(s) + dgr(r) \right]^2 \\ HM(G) &= \sum\_{sr \in E\_1} \left[ dgr(s) + dgr(r) \right]^2 + \sum\_{sr \in E\_2} \left[ dgr(s) + dgr(r) \right]^2 \\ &+ \sum\_{sr \in E\_3} \left[ dgr(s) + dgr(r) \right]^2 + \sum\_{sr \in E\_4} \left[ dgr(s) + dgr(r) \right]^2 \\ &= 14^2 |E\_1| + 16^2 |E\_2| + 17^2 |E\_3| + 18^2 |E\_4| \\ &= 196(4p) + 256(2p) + 289(4p) + 324(6pq - 5p) \\ &= 1944pq + 832p \end{aligned}$$

### • **Multiple Zagreb indices of** *GTUC***[***p***,** *q***], (***p***,** *q ≥* **1) Nanotube**

Let *G* = *GTUC*[*p*, *q*]. Now using Equations (2) and (3) we have

*PM*1(*G*) = ∏ *sr*∈*<sup>E</sup>*(*G*) [*dgr*(*s*) + *dgr*(*r*)] *PM*1*G* = ∏ *sr*∈*E*1 *dgr*(*s*) + *dgr*(*r*) × ∏ *sr*∈*E*2 *dgr*(*s*) + *dgr*(*r*) × ∏ *sr*∈*E*3 *dgr*(*s*) + *dgr*(*r*) × ∏ *sr*∈*E*4 *dgr*(*s*) + *dgr*(*r*) = 14|*<sup>E</sup>*1| × 16|*<sup>E</sup>*2| × 17|*<sup>E</sup>*3| × 18|*<sup>E</sup>*4| = 14(<sup>4</sup>*p*) × 16(<sup>2</sup>*p*) × 17(<sup>4</sup>*p*) × <sup>18</sup>(<sup>6</sup>*pq*−5*p*) *PM*2(*G*) = ∏ *sr*∈*<sup>E</sup>*(*G*) [*dgr*(*s*) × *dgr*(*r*)] *PM*2*G* = ∏ *sr*∈*E*1 *dgr*(*s*) × *dgr*(*r*) × ∏ *sr*∈*E*2 *dgr*(*s*) × *dgr*(*r*) × ∏ *sr*∈*E*3 *dgr*(*s*) × *dgr*(*r*) × ∏ *sr*∈*E*4 *dgr*(*s*) × *dgr*(*r*) = 48|*<sup>E</sup>*1| × 64|*<sup>E</sup>*2| × 72|*<sup>E</sup>*3| × 81|*<sup>E</sup>*4| = 48(<sup>4</sup>*p*) × 64(<sup>2</sup>*p*) × 72(<sup>4</sup>*p*) × <sup>81</sup>(<sup>6</sup>*pq*−5*p*)

### • **Zagreb polynomials of** *GTUC***[***p***,** *q***], (***p***,** *q ≥* **1) Nanotube**

Let *G* = *GTUC*[*p*, *q*]. Now using Equations (4) and (5) we have

$$\begin{array}{rcl}M\_{1}(G,\mathbf{x})&=&\sum\_{sr\in E(G)}\mathbf{x}^{[d\,\mathrm{gr}(s)+d\,\mathrm{gr}(r)]}\\M\_{1}(G,\mathbf{x})&=&\sum\_{sr\in E\_{1}}\mathbf{x}^{[d\,\mathrm{gr}(s)+d\,\mathrm{gr}(r)]}+\sum\_{sr\in E\_{2}}\mathbf{x}^{[d\,\mathrm{gr}(s)+d\,\mathrm{gr}(r)]}\\&+&\sum\_{sr\in E\_{3}}\mathbf{x}^{[d\,\mathrm{gr}(s)+d\,\mathrm{gr}(r)]}+\sum\_{sr\in E\_{4}}\mathbf{x}^{[d\,\mathrm{gr}(s)+d\,\mathrm{gr}(r)]}\\&=&\sum\_{sr\in E\_{1}}\mathbf{x}^{14}+\sum\_{sr\in E\_{2}}\mathbf{x}^{16}+\sum\_{sr\in E\_{3}}\mathbf{x}^{17}+\sum\_{sr\in E\_{4}}\mathbf{x}^{18}\end{array}$$

$$\begin{array}{rcl}M\_{1}(\mathcal{G},\mathbf{x})&=&|E\_{1}|\mathbf{x}^{14}+|E\_{2}|\mathbf{x}^{16}+|E\_{3}|\mathbf{x}^{17}+|E\_{4}|\mathbf{x}^{18}\\&=&(4p)\mathbf{x}^{14}+(2p)\mathbf{x}^{16}+(4p)\mathbf{x}^{17}+(6pq-5p)\mathbf{x}^{18}\\M\_{2}(\mathcal{G},\mathbf{x})&=&\sum\_{\mathbf{x}\in E(\mathcal{G})}\mathbf{x}^{[\deg\tau(\mathbf{s})\times\deg\tau(\mathbf{r})]}\\M\_{2}(\mathcal{G},\mathbf{x})&=&\sum\_{\mathbf{s}\in E\_{1}}\mathbf{x}^{[\deg\tau(\mathbf{s})\times\deg\tau(\mathbf{r})]}+\sum\_{\mathbf{s}\in E\_{2}}\mathbf{x}^{[\deg\tau(\mathbf{s})\times\deg\tau(\mathbf{r})]}\\&+&\sum\_{\mathbf{s}\in E\_{3}}\mathbf{x}^{[\deg\tau(\mathbf{s})\times\deg\tau(\mathbf{r})]}+\sum\_{\mathbf{s}\in E\_{4}}\mathbf{x}^{[\deg\tau(\mathbf{s})\times\deg\tau(\mathbf{r})]}\\&=&\sum\_{\mathbf{s}\in E\_{1}}\mathbf{x}^{48}+\sum\_{\mathbf{s}\in\mathcal{E}\_{2}}\mathbf{x}^{64}+\sum\_{\mathbf{s}\in E\_{3}}\mathbf{x}^{72}+\sum\_{\mathbf{s}\in E\_{4}}\mathbf{x}^{81}\\&=&|E\_{1}|\mathbf{x}^{48}+|E\_{2}|\mathbf{x}^{64}+|E\_{3}|\mathbf{x}^{72}+|E\_{4}|\mathbf{x}^{81}\\&=&(4p)\mathbf{x}^{48}+(2p)\mathbf{x}^{64}+(4p)\mathbf{x}^{72}+(6pq-5p)\mathbf{x}^{81}\\&\end{array}$$
