*Drawing Algorithm of Third Type of Hex-Derived Networks HDN*<sup>3</sup>

Step-1: For *HDN*3, we should draw a hexagonal mesh of dimension *m*.

Step-2: Draw a *K*<sup>3</sup> graph in each subgraph of *K*<sup>3</sup> and join all the vertices to the outer vertices of each *K*3. The new graph is called an *HDN*3 (see Figure 2) network.

Step-3: By *HDN*<sup>3</sup> network, we can simply design *THDN*<sup>3</sup> (see Figure 3) and *RHDN*<sup>3</sup> (see Figure 4).

**Figure 1.** Hexagonal meshes: (1) HX2, (2) HX3, and (3), all facing HX2.

**Figure 2.** Third type of hex-derived network (*HDN*3(4)).

In this paper, '*ξ*' is taken as a simple connected graph and the degree of any vertex *m*´ ∈ *V*(*ξ*) is stands for *τ*(*m*´).

The oldest, most desired and supremely studied degree-based topological index was introduced by Milan Randi´c and is known as *Randi´c index* [2] denoted by *<sup>R</sup>*<sup>−</sup> <sup>1</sup> 2 (*ξ*) and described as

$$\mathcal{R}\_{-\frac{1}{2}}(\xi) = \sum\_{\vec{m}\vec{n}\in\mathcal{E}(\xi)} \frac{1}{\sqrt{\pi(\vec{m})\pi(\vec{n})}}.\tag{1}$$

The *Forgotten index*, also called F-index, was discovered by Furtula and Ivan Gutman [3] and described as

$$F(\xi) = \sum\_{\mathfrak{m}\mathfrak{m} \in \mathbb{E}(\xi)} (\left(\tau(\mathfrak{m})\right)^2 + \left(\tau(\mathfrak{m})\right)^2). \tag{2}$$

**Figure 3.** Third type of triangular hex-derived network (*THDN*3(7)).

**Figure 4.** Third type of rectangular hex-derived network (*RHDN*3(4, 4)).

In 1982, Balaban [4,5] found another important index known as *Balaban index*. For a graph *ξ* of '*n*' vertices and '*m*' edges, and is described as

$$J(\xi) = \left(\frac{m}{m - n + 2}\right) \sum\_{\mathfrak{n} \models \mathfrak{n} \in \mathbb{E}(\xi)} \frac{1}{\sqrt{\pi(\mathfrak{n}) \times \pi(\mathfrak{n})}}.\tag{3}$$

The reclassified the Zagreb indices which are proposed by Ranjini et al. [6], is of three types. For a graph *ξ*, it is described as

$$R\varepsilon Z G\_1(\vec{\xi}) = \sum\_{\mathfrak{m}\mathfrak{m} \in \mathcal{E}(\vec{\xi})} \left( \frac{\mathfrak{r}(\vec{m}) \times \mathfrak{r}(\vec{\alpha})}{\mathfrak{r}(\vec{m}) + \mathfrak{r}(\vec{\alpha})} \right),\tag{4}$$

$$\text{ReZG}\_2(\xi) = \sum\_{\mathfrak{m}\mathfrak{m} \in \text{E}(\xi)} \left( \frac{\mathfrak{r}(\acute{m}) + \mathfrak{r}(\acute{n})}{\mathfrak{r}(\acute{m}) \times \mathfrak{r}(\acute{n})} \right),\tag{5}$$

$$\operatorname{ReZG}\_3(\xi) = \sum\_{\vec{m}\vec{n} \in \mathcal{E}(\xi)} (\pi(\mathfrak{m}) \times \pi(\mathfrak{n})) (\pi(\mathfrak{m}) + \pi(\mathfrak{n})).\tag{6}$$

The atom-bond connectivity (ABC) index is a useful predictive index in the study of the heat of formation in alkanes [7] and is introduced by Estrada et al. [8].

Ghorbani et al. [9] introduced the *ABC*<sup>4</sup> index and is described as

$$ABC\_4(\vec{\xi}) = \sum\_{\mathfrak{m}\mathfrak{h} \in \mathcal{E}(\vec{\xi})} \sqrt{\frac{S\_{\mathfrak{M}} + S\_{\mathfrak{H}} - 2}{S\_{\mathfrak{M}} S\_{\mathfrak{H}}}}.\tag{7}$$

Graovac et al. [10] introduced the *GA*<sup>5</sup> index and is described as

$$GA\_{\mathfrak{F}}(\xi) = \sum\_{\mathfrak{M}\mathfrak{h} \in \mathbb{E}(\xi)} \frac{2\sqrt{S\_{\mathfrak{M}}S\_{\mathfrak{H}}}}{(S\_{\mathfrak{M}} + S\_{\mathfrak{H}})}.\tag{8}$$

#### **2. Main Results**

Simonraj et al. [11] created the new network which is named as third type of hex-derived networks. Chang-Cheng Wei et al. [12] found some topological indices of certain new derived networks. In this paper, we compute the exact results for all the above descriptors. For these results on different degree-based topological descriptors for a variety of graphs, we recommend [13–20]. For the basic notations and definitions, see [21,22].

### *2.1. Results for HDN*3(*m*)

In this part, the Forgotten index, Balaban index, reclassified the Zagreb indices, *ABC*<sup>4</sup> index, and *GA*<sup>5</sup> index are under consideration for the third type of hex-derived network.

**Theorem 1.** *Consider the third type of hex-derived network HDN*3(*m*)*; its Forgotten index is equal to*

$$F(HDN\_3(m)) = 6(5339 - 8132n + 3108n^2).$$

**Proof.** Let *ξ*<sup>1</sup> be the hex-derived network of Type 3, *HDN*3(*m*) shown in Figure 2, where *m* ≥ 4. The hex derived network *<sup>ξ</sup>*<sup>1</sup> has 21*m*<sup>2</sup> <sup>−</sup> <sup>39</sup>*<sup>m</sup>* <sup>+</sup> 19 vertices and the edge set of *<sup>ξ</sup>*<sup>1</sup> is divided into nine partitions based on the degrees of end vertices as shown in Table 1.

Forgotten index can be calculated by using Table 1. Thus, from (2), it follows,

$$\begin{array}{rcl} F(\xi\_1) &=& 32|E\_1(\xi\_1)| + 65|E\_2(\xi\_1)| + 116|E\_3(\xi\_1)| + 340|E\_4(\xi\_1)| + 149|E\_5(\xi\_1)| + 373|E\_6(\xi\_1)| + \\ &200|E\_7(\xi\_1)| + 424|E\_8(\xi\_1)| + 648|E\_9(\xi\_1)|. \end{array}$$

After some calculations, we have the final result

$$\implies F(\xi\_1) = 6(5339 - 8132n + 3108n^2).$$

**Table 1.** Edge partition of third type of hex-derived network *HDN*3(*m*), based on degrees of end vertices of each edge.


In the subsequent theorem, we compute the Balaban index of the third type of hex-derived network, *ξ*1.

**Theorem 2.** *For the third type of hex-derived network ξ*1*, the Balaban index is equal to*

$$J(\xi\_1) = \begin{pmatrix} 1 \\ 70(43 - 84m + 42m^2) \end{pmatrix} ((20 - 41m + 21m^2)(1595.47 + 7(-307 - 270\sqrt{2} + 12\sqrt{5} + 35))$$
  $54\sqrt{10}(m) + 210(5 + 3\sqrt{2})m^2$ )

**Proof.** Let *ξ*<sup>1</sup> be the third type of hex-derived network *HDN*3(*m*). The Balaban index can be calculated by using (3) and with the help of Table 1, we have.

$$\begin{split} J(\boldsymbol{\xi}\_{1}) &= \quad \left( \frac{63n^{2} - 123n + 60}{43 - 84n + 42n^{2}} \right) \Big( \frac{1}{4} |\boldsymbol{E}\_{1}(\boldsymbol{\xi}\_{1})| + \frac{1}{2\sqrt{7}} |\boldsymbol{E}\_{2}(\boldsymbol{\xi}\_{1})| + \frac{1}{2\sqrt{10}} |\boldsymbol{E}\_{3}(\boldsymbol{\xi}\_{1})| + \frac{1}{6\sqrt{2}} |\boldsymbol{E}\_{4}(\boldsymbol{\xi}\_{1})| + \dots \\ & \quad \frac{1}{\sqrt{70}} |\boldsymbol{E}\_{5}(\boldsymbol{\xi}\_{1})| + \frac{1}{3\sqrt{14}} |\boldsymbol{E}\_{6}(\boldsymbol{\xi}\_{1})| + \frac{1}{10} |\boldsymbol{E}\_{7}(\boldsymbol{\xi}\_{1})| + \frac{1}{6\sqrt{5}} |\boldsymbol{E}\_{8}(\boldsymbol{\xi}\_{1})| + \frac{1}{18} |\boldsymbol{E}\_{9}(\boldsymbol{\xi}\_{1})| \Big). \end{split}$$

After some calculations, we have the result

$$\begin{array}{rcl} \implies J(\mathbf{\bar{z}}\_1) &=& \left(\frac{1}{70(43 - 84m + 42m^2)}\right)((20 - 41m + 21m^2)(1595.47 + 7(-307 - 270\sqrt{2} + 12\sqrt{5} + 36)) \\ &+& 54\sqrt{10})m) + 210(5 + 3\sqrt{2})m^2). \end{array}$$

Now, we compute *ReZG*1, *ReZG*<sup>2</sup> and *ReZG*<sup>3</sup> indices of the third type of hex-derived network *ξ*1.

**Theorem 3.** *Let ξ*<sup>1</sup> *be the third type of hex-derived network, then*


**Proof.** Reclassified Zagreb index can be calculated by using Table 1, the ReZ*G*1(*ξ*1) by using Equation (4) as follows.

$$\begin{split} \operatorname{Re} Z G\_{1}(\underline{\mathcal{F}}\_{1}) &= 2|E\_{1}(\underline{\mathcal{F}}\_{1})| + \frac{28}{11}|E\_{2}(\underline{\mathcal{F}}\_{1})| + \frac{20}{7}|E\_{3}(\underline{\mathcal{F}}\_{1})| + \frac{36}{11}|E\_{4}(\underline{\mathcal{F}}\_{1})| + \frac{70}{17}|E\_{5}(\underline{\mathcal{F}}\_{1})| + \frac{126}{25}|E\_{6}(\underline{\mathcal{F}}\_{1})| + \frac{20}{7}|E\_{7}(\underline{\mathcal{F}}\_{1})| \\ &\leq |E\_{7}(\underline{\mathcal{F}}\_{1})| + \frac{45}{7}|E\_{8}(\underline{\mathcal{F}}\_{1})| + 9|E\_{9}(\underline{\mathcal{F}}\_{1})|. \end{split}$$

After some calculations, we have

$$\implies \text{Re}ZG\_1(\xi\_1) = 19 - 39m + 21m^2.$$

The ReZG2(*ξ*1) can be calculated by using (5) as follows.

$$\begin{array}{rcl} ReZG\_2(\xi\_1) &=& \frac{1}{2}|E\_1(\xi\_1)| + \frac{11}{28}|E\_2(\xi\_1)| + \frac{7}{20}|E\_3(\xi\_1)| + \frac{11}{36}|E\_4(\xi\_1)| + \frac{17}{70}|E\_5(\xi\_1)| + \frac{11}{9}|E\_6(\xi\_1)|\\ & \frac{25}{126}|E\_6(\xi\_1)| + \frac{1}{5}|E\_7(\xi\_1)| + \frac{7}{45}|E\_8(\xi\_1)| + \frac{1}{9}|E\_9(\xi\_1)|.\\ \end{array}$$

After some calculations, we have

$$\implies \text{Re}Z G\_2(\xi\_1) = \frac{115452}{425} - \frac{5637m}{11} + \frac{2583m^2}{11}.$$

The ReZG3(*ξ*1) index can be calculated from (6) as follows.

$$\operatorname{ReZG}\_3(\mathbb{f}\_1) \;= \sum\_{\substack{\pi \mathsf{i}\mathsf{i}\mathsf{i}\in\mathsf{E}(\mathbb{f}\_1)}} (\pi(\mathsf{i}\mathsf{i}) \times \pi(\mathsf{i}\mathsf{i}))(\pi(\mathsf{i}\mathsf{i}) + \pi(\mathsf{i}\mathsf{i}) = \sum\_{\substack{\pi \mathsf{i}\mathsf{i}\in\mathsf{E}\_j(\mathbb{f}\_1)}} \sum\_{j=1}^9 (\pi(\mathsf{i}\mathsf{i}) \times \pi(\mathsf{i}\mathsf{i}))(\pi(\mathsf{i}\mathsf{i}) + \pi(\mathsf{i}\mathsf{i})) $$

$$\begin{array}{rcl} \text{Re}Z G \mathfrak{z} \{ \mathfrak{z}\_1 \} &=& 128 \left| E\_1(\mathfrak{z}\_1) \right| + 308 \left| E\_2(\mathfrak{z}\_1) \right| + 560 \left| E\_3(\mathfrak{z}\_1) \right| + 1584 \left| E\_4(\mathfrak{z}\_1) \right| + 1190 \left| E\_5(\mathfrak{z}\_1) \right| + 1\\ & & \\ & 3150 \left| E\_6(\mathfrak{z}\_1) \right| + 2000 \left| E\_7(\mathfrak{z}\_1) \right| + 5040 \left| E\_8(\mathfrak{z}\_1) \right| + 11664 \left| E\_9(\mathfrak{z}\_1) \right|. \end{array}$$

After some calculations, we have

$$\implies \text{Re}ZG\_3(\xi\_1) \; = \; \; 12(27381 - 38996m + 13692m^2) \; .$$

Now, we find *ABC*<sup>4</sup> and *GA*<sup>5</sup> indices of third type of hex-derived network *ξ*1.

**Theorem 4.** *Let ξ*<sup>1</sup> *be the third type of hex-derived network, then*


**Proof.** The *ABC*4(*ξ*1) index can be calculated by using (7) and by Table 2, as follows.

$$\begin{split} ABC\_{4}(\xi\_{1}) &= \quad \frac{2}{5} \sqrt{\frac{14}{33}} |E\_{10}(\xi\_{1})| + \frac{\sqrt{59}}{30} |E\_{11}(\xi\_{1})| + \frac{1}{15} \sqrt{\frac{77}{6}} |E\_{12}(\xi\_{1})| + \frac{36}{11} \frac{2}{\sqrt{77}} |E\_{13}(\xi\_{1})| + \\ & \quad \frac{1}{6} \sqrt{\frac{31}{14}} |E\_{14}(\xi\_{1})| + \frac{1}{14} \sqrt{\frac{103}{11}} |E\_{15}(\xi\_{1})| + \frac{1}{4} \sqrt{\frac{53}{70}} |E\_{16}(\xi\_{1})| + \frac{1}{6} \sqrt{\frac{67}{33}} |E\_{17}(\xi\_{1})| + \\ & \quad \frac{1}{9} \sqrt{\frac{85}{22}} |E\_{18}(\xi\_{1})| + \frac{4}{3} \sqrt{\frac{10}{473}} |E\_{19}(\xi\_{1})| + \frac{1}{18} \sqrt{\frac{32}{21}} |E\_{20}(\xi\_{1})| + \frac{1}{2} \sqrt{\frac{13}{66}} |E\_{21}(\xi\_{1})| + \\ & \quad \frac{1}{2} \sqrt{\frac{37}{231}} |E\_{22}(\xi\_{1})| + \frac{1}{4} \sqrt{\frac{19}{30}} |E\_{23}(\xi\_{1})| + \frac{1}{6} \sqrt{\frac{163}{129}} |E\_{24}(\xi\_{1})| + \frac{1}{2} \sqrt{\frac{29}{101}} |E\_{25}(\xi\_{1})| + \\ & \quad \frac{1}{22} \sqrt{\frac{43}{2$$

After some calculations, we have

$$\begin{array}{rcl} \implies \text{ABC}\_4(\xi\_1) &=& 51.706 + \frac{3}{20} \sqrt{\frac{79}{2}}(-5+m) + 3\sqrt{\frac{53}{70}}(-4+m) + \frac{3}{5} \sqrt{\frac{109}{14}}(-4+m) + \\ & & \sqrt{\frac{114}{5}}(-4+m) + \frac{3}{35} \sqrt{\frac{139}{2}}(-4+m) + 3\sqrt{\frac{14}{65}}(-3+m) + 12\sqrt{\frac{26}{55}}(-3+m) + 3\sqrt{\frac{17}{100}}(-3+m) + 12\sqrt{\frac{17}{100}}(-3+m) + 12\sqrt{\frac{17}{100}}(-2+m) + 12\sqrt{\frac{17}{100}}(-2+m) + 12\sqrt{\frac{17}{100}}(-2+m) + 12\sqrt{\frac{17}{100}}(-2+m) + 12\sqrt{\frac{17}{100}}(-2+m) + 12\sqrt{\frac{17}{100}}(-2+m) + 12\sqrt{\frac{17}{100}}(-2+m) + 12\sqrt{\frac{17}{100}}(-2+m) + 12\sqrt{\frac{17}{100}}(-2+m) + 12\sqrt{\frac{17}{100}}(-2+m) + 12\sqrt{\frac{17}{100}}(-2+m) + 12\sqrt{\frac{17}{100}}(-2+m) + 12\sqrt{\frac{17}{100}}(-2+m) + 12\sqrt{\frac{17}{100}}(-2+m) + 12\sqrt{\frac{17}{100}}(-2+m) + 12\sqrt{\frac{17}{100}}(-2+m) + 12\sqrt{\frac{17}{100}}(-2+m) + 12\sqrt{\frac{17}{100}}(-2+m) \end{array}$$

The *GA*5(*ξ*1) index can be determined from (8) as follows.

*GA*5(*ξ*1) = <sup>5</sup> √ |*E*10(*ξ*1)| + <sup>|</sup>*E*11(*ξ*1)<sup>|</sup> <sup>+</sup> √ |*E*12(*ξ*1)| + √ |*E*13(*ξ*1)| + √ |*E*14(*ξ*1)| + √ |*E*15(*ξ*1)| + √ |*E*16(*ξ*1)| + √ |*E*17(*ξ*1)| + √ |*E*18(*ξ*1)| + √ |*E*19(*ξ*1)| + |*E*20(*ξ*1)| + √ |*E*21(*ξ*1)| + √ |*E*22(*ξ*1)| + √ |*E*23(*ξ*1)| + √ |*E*24(*ξ*1)| + √ |*E*25(*ξ*1)| + |*E*26(*ξ*1)| + √ |*E*27(*ξ*1)| + √ |*E*28(*ξ*1)| + √ |*E*29(*ξ*1)| + √ |*E*30(*ξ*1)| + √ |*E*31(*ξ*1)| + √ |*E*32(*ξ*1)| + √ |*E*33(*ξ*1)| + √ |*E*34(*ξ*1)<sup>|</sup> <sup>+</sup> <sup>|</sup>*E*35(*ξ*1) + <sup>4</sup> √ |*E*36(*ξ*1)| + √ |*E*37(*ξ*1)| + √ |*E*38(*ξ*1)| + |*E*39(*ξ*1)| + √ |*E*40(*ξ*1)| + |*E*41(*ξ*1)|.

After some calculations, we have

=⇒ *GA*5(*ξ*1) = 315.338 + √ (−<sup>4</sup> <sup>+</sup> *<sup>m</sup>*) + <sup>48</sup> √ (−<sup>4</sup> <sup>+</sup> *<sup>m</sup>*) + <sup>16</sup> √ (−<sup>4</sup> <sup>+</sup> *<sup>m</sup>*) + <sup>9</sup> √ (−<sup>3</sup> <sup>+</sup> *<sup>m</sup>*) + <sup>36</sup> √ (−<sup>3</sup> <sup>+</sup> *<sup>m</sup>*) + <sup>48</sup> √ (−<sup>3</sup> <sup>+</sup> *<sup>m</sup>*) + <sup>12</sup> √ (−3 + *m*) + √ (−<sup>2</sup> <sup>+</sup> *<sup>m</sup>*) <sup>−</sup> <sup>99</sup>*<sup>m</sup>* <sup>+</sup> <sup>27</sup>*m*<sup>2</sup> <sup>+</sup> √ (<sup>19</sup> <sup>−</sup> <sup>15</sup>*<sup>m</sup>* <sup>+</sup> <sup>3</sup>*m*2).

**Table 2.** Edge partition of the third type of hex-derived network *HDN*3(*m*) based on sum of degrees of end vertices of each edge.


*2.2. Results for Third Type of Triangular Hex-Derived Network THDN*3(*m*)

Now, we discuss the third type of rectangular hex-derived network and compute exact results for Forgotten index and Balaban index, and reclassified the Zagreb indices, *ABC*<sup>4</sup> index, and *GA*<sup>5</sup> index for *THDN*3(*m*).

**Theorem 5.** *Consider the third type of triangular hex-derived network of THDN*3(*n*)*; its Forgotten index is equal to*

$$F(THDN\_3(m)) = 12(990 - 997m + 259m^2).$$

**Proof.** Let *ξ*<sup>2</sup> be the third type of triangular hex-derived network, *THDN*3(*m*) shown in Figure 3, where *<sup>m</sup>* <sup>≥</sup> 4. The third type of triangular hex-derived network *<sup>ξ</sup>*<sup>2</sup> has <sup>7</sup>*m*2−11*m*+<sup>6</sup> <sup>2</sup> vertices and the edge set of *ξ*<sup>2</sup> is divided into six partitions based on the degree of end vertices as shown in Table 3.

By using edge partition from Table 3, we get. Thus, from (2) it follows that

*F*(*ξ*2) = 32|*E*1(*ξ*2)| + 116|*E*2(*ξ*2)| + 340|*E*3(*ξ*2)| + 200|*E*4(*ξ*2)| + 424|*E*5(*ξ*2)| + 648|*E*6(*ξ*2)|.

By doing some calculations, we get

$$\implies F(\xi\_2) = 12(990 - 997m + 259m^2).$$

**Table 3.** Edge partition of the third type of triangular hex-derived network *THDN*3(*m*) based on degrees of end vertices of each edge.


In the following theorem, we compute the Balaban index of the third type of triangular hex-derived network, *ξ*2.

**Theorem 6.** *For the third type of triangular hex-derived network ξ*2*, the Balaban index is equal to*

*<sup>J</sup>*(*ξ*2) = <sup>1</sup> <sup>40</sup>(<sup>8</sup> − <sup>14</sup>*<sup>m</sup>* + <sup>7</sup>*m*2) <sup>6</sup> <sup>−</sup> <sup>13</sup>*<sup>m</sup>* <sup>+</sup> <sup>7</sup>*m*2)(<sup>159</sup> <sup>+</sup> <sup>1802</sup><sup>√</sup> <sup>2</sup> <sup>−</sup> <sup>36</sup><sup>√</sup> <sup>5</sup> <sup>−</sup> <sup>90</sup><sup>√</sup> <sup>10</sup> + (−<sup>107</sup> <sup>−</sup> <sup>150</sup><sup>√</sup> 2 + 12<sup>√</sup> <sup>5</sup> <sup>+</sup> <sup>54</sup><sup>√</sup> 10)*m* + 10(5 + 3 √ 2)*m*<sup>2</sup> .

**Proof.** Let *ξ*<sup>2</sup> be the third type of triangular hex-derived network *THDN*3(*m*). By using edge partition from Table 3, the result follows. The Balaban index can be calculated by using (3) as follows.

$$\begin{split} f(\xi\_2) &= \quad \frac{3}{2} \Big( \frac{6 - 13m + 7m^2}{8 - 14m + 7m^2} \Big) \Big( \frac{1}{4} |E\_1(\xi\_2)| + \frac{1}{2\sqrt{10}} |E\_2(\xi\_2)| + \frac{1}{6\sqrt{2}} |E\_3(\xi\_2)| + \frac{1}{10} |E\_4(\xi\_2)| + \frac{1}{2} |E\_5(\xi\_2)| + \frac{1}{10} |E\_6(\xi\_2)| + \frac{1}{10} |E\_7(\xi\_2)| \Big) \\ &\quad \frac{1}{6\sqrt{5}} |E\_5(\xi\_2)| + \frac{1}{18} |E\_6(\xi\_2)| \Big). \end{split}$$

After some calculation, we have

$$\implies f(\vec{z}\_2) \quad = \begin{pmatrix} 1 \\ \frac{1}{40(8 - 14m + 7m^2)} \end{pmatrix} \Big( 6 - 13m + 7m^2 \Big) (159 + 1802\sqrt{2} - 36\sqrt{5} - 90\sqrt{10} + (-107 - 90\sqrt{5}) + 36\sqrt{10})$$

$$150\sqrt{2} + 12\sqrt{5} + 54\sqrt{10})m + 10(5 + 3\sqrt{2})m^2 \Big).$$

Now, we compute *ReZG*1, *ReZG*<sup>2</sup> and *ReZG*<sup>3</sup> indices of third type of triangular hex-derived network *ξ*2.

**Theorem 7.** *Let ξ*<sup>2</sup> *be the third type of triangular hex-derived network, then*


**Proof.** By using edge partition given in Table 3, the ReZG1(*ξ*2) can be calculated by using (4) as follows.

$$\begin{array}{rcl} \text{Re}\mathcal{Z}\mathcal{G}\_{1}(\xi\_{2}) &=& 2|E\_{1}(\xi\_{2})| + \frac{20}{7}|E\_{2}(\xi\_{2})| + \frac{36}{11}|E\_{3}(\xi\_{2})| + 5|E\_{4}(\xi\_{2})| + \frac{45}{7}|E\_{5}(\xi\_{2})| + 9|E\_{6}(\xi\_{2})|. \end{array}$$

After some calculation, we have

$$\implies \text{Re}Z G\_1(\xi\_2) = \frac{3}{154}(3408 - 5117m + 2009m^2).$$

The ReZG2(*ξ*2) can be calculated by using (5) as follows.

$$\text{Re}Z G\_2(\xi\_2) = \frac{1}{2}|E\_1(\xi\_2)| + \frac{7}{20}|E\_2(\xi\_2)| + \frac{11}{36}|E\_3(\xi\_2)| + \frac{1}{5}|E\_4(\xi\_2)| + \frac{7}{45}|E\_5(\xi\_2)| + \frac{1}{9}|E\_6(\xi\_2)|.$$

After some calculation, we have

$$\implies \text{Re}ZG\_2(\xi\_2) = \frac{1}{2}(6 - 11m + 7m^2).$$

The ReZG3(*ξ*2) index can be calculated from (6) as follows.

$$\begin{array}{rcl} \text{Re}Z G\_3(\not\!\!\!\_2) &=& 128|E\_1(\not\!\!\!\_2)| + 560|E\_2(\not\!\!\!\_2)| + 1584|E\_3(\not\!\!\!\_2)| + 2000|E\_4(\not\!\!\!\_2)| + 5040|E\_5(\not\!\!\!\!\!\_2)| + 158|E\_6(\not\!\!\!\!\!\!\!\_2)| \\ &=& 11664|E\_6(\not\!\!\!\!\_2)|. \end{array}$$

After some calculation, we have

$$\implies \text{Re}ZG\_3(\xi\_2) \quad = \ 24(6192 - 5185m + 1141m^2).$$

Now, we compute *ABC*<sup>4</sup> and *GA*<sup>5</sup> indices of third type of triangular hex-derived network *ξ*2.

**Theorem 8.** *Let ξ*<sup>2</sup> *be the third type of triangular hex-derived network, then*


**Proof.** By using the edge partition given in Table 4, the *ABC*4(*ξ*2) index can be calculated by using (7) as follows.

*ABC*4(*ξ*2) = <sup>1</sup> <sup>21</sup> <sup>|</sup>*E*7(*ξ*2)<sup>|</sup> <sup>+</sup> 6 <sup>|</sup>*E*8(*ξ*2)<sup>|</sup> <sup>+</sup> 7 <sup>|</sup>*E*9(*ξ*2)<sup>|</sup> <sup>+</sup> <sup>43</sup> <sup>|</sup>*E*10(*ξ*2)<sup>|</sup> <sup>+</sup> 23 <sup>|</sup>*E*11(*ξ*2)<sup>|</sup> <sup>+</sup> <sup>|</sup>*E*12(*ξ*2)<sup>|</sup> <sup>+</sup> <sup>32</sup> <sup>|</sup>*E*13(*ξ*2)<sup>|</sup> <sup>+</sup> <sup>|</sup>*E*14(*ξ*2)<sup>|</sup> <sup>+</sup> √6 |*E*15(*ξ*2)| + <sup>|</sup>*E*16(*ξ*2)<sup>|</sup> <sup>+</sup> <sup>|</sup>*E*17(*ξ*2)<sup>|</sup> <sup>+</sup> 29 <sup>|</sup>*E*18(*ξ*2)<sup>|</sup> <sup>+</sup> <sup>43</sup> <sup>|</sup>*E*19(*ξ*2)<sup>|</sup> <sup>+</sup> 83 <sup>|</sup>*E*20(*ξ*2)<sup>|</sup> <sup>+</sup> 13 <sup>|</sup>*E*21(*ξ*2)<sup>|</sup> <sup>+</sup> 3 <sup>|</sup>*E*22(*ξ*2)<sup>|</sup> <sup>+</sup> <sup>65</sup> <sup>|</sup>*E*23(*ξ*2)<sup>|</sup> <sup>+</sup> 3 <sup>|</sup>*E*24(*ξ*2)<sup>|</sup> <sup>+</sup> 47 <sup>|</sup>*E*25(*ξ*2)<sup>|</sup> <sup>+</sup> <sup>79</sup> <sup>|</sup>*E*26(*ξ*2)<sup>|</sup> <sup>+</sup> <sup>|</sup>*E*27(*ξ*2)<sup>|</sup> <sup>+</sup> <sup>109</sup> <sup>|</sup>*E*28(*ξ*2)<sup>|</sup> <sup>+</sup> 131 <sup>|</sup>*E*29(*ξ*2)<sup>|</sup> <sup>+</sup> <sup>139</sup> <sup>|</sup>*E*30(*ξ*2)<sup>|</sup> <sup>+</sup> 7 <sup>|</sup>*E*31(*ξ*2)<sup>|</sup> <sup>+</sup> <sup>155</sup> <sup>|</sup>*E*32(*ξ*2)|.

After some calculation, we have

$$\begin{array}{rcl} \implies \text{ABC}\_4(\xi\_2) &=& 24.131 + 3\sqrt{\frac{7}{130}}(-6+m) + 6\sqrt{\frac{26}{55}}(-5+m) + \sqrt{\frac{174}{35}}(-5+m) + \\ & \frac{3}{10}\sqrt{\frac{109}{14}}(-5+m) + \frac{3}{40}\sqrt{\frac{79}{2}}(-5+m) + \frac{3}{70}\sqrt{\frac{139}{2}}(-5+m) + \frac{3}{2}\sqrt{\frac{53}{70}}(-4+m) + \\ & \sqrt{\frac{39}{22}}(-4+m) + \sqrt{\frac{57}{10}}(-4+m) + \frac{3}{22}\sqrt{\frac{43}{2}}(-4+m)^2 + \frac{1}{3}\sqrt{\frac{35}{2}}(-3+m) + \\ & 2\sqrt{\frac{7}{11}}(-2+m) + \frac{1}{52}\sqrt{\frac{155}{2}}(42-13m+m^2) + 3\sqrt{\frac{3}{26}}(30-11m+m^2). \end{array}$$

The *GA*5(*ξ*2) index can be calculated from (8) as follows.

*GA*5(*ξ*2) = 1|*E*7(*ξ*2)| + √ |*E*8(*ξ*2)| + √ |*E*9(*ξ*2)| + √ |*E*10(*ξ*2)| + √ |*E*11(*ξ*2)| + √ |*E*12(*ξ*2)| + 1|*E*13(*ξ*2)| + √ |*E*14(*ξ*2)| + √ |*E*15(*ξ*2)| + √ |*E*16(*ξ*2)| + √ |*E*17(*ξ*2)| + √ |*E*18(*ξ*2)| + |*E*19(*ξ*2)| + √ |*E*20(*ξ*2)| + √ |*E*21(*ξ*2)| + √ |*E*22(*ξ*2)| + |*E*23(*ξ*2)| + √ |*E*24(*ξ*2)| + √ |*E*25(*ξ*2)| + 1|*E*26(*ξ*2)| + √ |*E*27(*ξ*2)| + √ |*E*28(*ξ*2)| + √ |*E*29(*ξ*2)| + 1|*E*30(*ξ*2)| + √ |*E*31(*ξ*2)| + 1|*E*32(*ξ*2)|.

After some calculation, we have

$$\begin{split} \implies \square G A\_5(\tilde{\xi}\_2) &= \ 110.66 + \frac{6}{37} \sqrt{1365}(-6 + m) + \frac{24}{11} \sqrt{7}(-5 + m) + \frac{18}{11} \sqrt{35}(-5 + m) + \frac{24}{23} \sqrt{385}(-5 + m) \\ &\quad \quad m) + \frac{144}{29} \sqrt{5}(-4 + m) + \frac{9}{5} \sqrt{11}(-4 + m) + \frac{8}{9} \sqrt{35}(-4 + m) + \frac{36}{29} \sqrt{22}(-2 + m) - 12m + \frac{17}{11} \sqrt{35}(-15 + m) + \frac{17}{11} \sqrt{35}(-2 + m) + \frac{17}{11} \sqrt{35}(-2 + m) - 12m + \frac{17}{11} \sqrt{35}(-2 + m) + \frac{17}{11} \sqrt{35}(-2 + m) + \frac{17}{11} \sqrt{35}(-2 + m) - 12m + \frac{17}{11} \sqrt{35}(-2 + m) - 12m + \frac{17}{11} \sqrt{35}(-2 + m) - 12m + \frac{17}{11} \sqrt{35}(-2 + m) - 12m + \frac{17}{11} \sqrt{35}(-2 + m) \end{split}$$


**Table 4.** Edge partition of the third type of triangular hex-derived network *THDN*3(*m*) based on the sum of degrees of end vertices of each edge.

*2.3. Results for Third Type of Rectangular Hex-Derived Network, RHDN*3(*m*, *n*)

In this section, we calculate certain degree-based topological indices of the third type of rectangular hex-derived network, *RHDN*3(*m*, *n*) of dimension *m* = *n*. We compute Forgotten index and Balaban index, and reclassified the Zagreb indices, forth version of *ABC* index, and fifth version of *GA* index in the coming theorems of *RHDN*3(*m*, *n*).

**Theorem 9.** *Consider the third type of rectangular hex-derived network RHDN*3(*m*)*, its Forgotten index is equal to*

$$F(RHDN\_3(m)) = 19726 - 20096m + 6216m^2.$$

**Proof.** Let *ξ*<sup>3</sup> be the third type of rectangular hex-derived network, *RHDN*3(*m*) shown in Figure 4, where *<sup>m</sup>* <sup>=</sup> *<sup>s</sup>* <sup>≥</sup> 4. The third type of rectangular hex-derived network *<sup>ξ</sup>*<sup>3</sup> has 7*m*<sup>2</sup> <sup>−</sup> <sup>12</sup>*<sup>m</sup>* <sup>+</sup> 6 vertices and the edge set of *ξ*<sup>3</sup> is divided into nine partitions based on the degree of end vertices as shown in Table 5.

**Table 5.** Edge partition of the third type of rectangular hex-derived network, *RHDN*3(*m*) based on degrees of end vertices of each edge.


Thus, from (2), it follows that.

$$F(G) = \sum\_{\mathfrak{n}\mathfrak{h}\mathfrak{h} \in \mathrm{E}(\mathfrak{f})} \left( (\mathfrak{r}(\mathfrak{m}))^2 + (\mathfrak{r}(\mathfrak{n}))^2 \right)^2$$

Let *ξ*<sup>3</sup> be the third type of rectangular hex-derived network, *THDN*3(*m*). By using edge partition from Table 5, the result follows.

$$F(\xi\_3) = \sum\_{\mathfrak{m}\mathfrak{n}\mathfrak{t} \in \mathbb{E}(\xi\_3)} ((\tau(\mathfrak{m}))^2 + (\tau(\mathfrak{n}))^2) = \sum\_{\mathfrak{m}\mathfrak{n}\mathfrak{t} \in \mathbb{E}\_{\mathfrak{l}}(\xi\_3)} \sum\_{j=1}^9 ((\tau(\mathfrak{m}))^2 + (\tau(\mathfrak{n}))^2).$$

$$\begin{array}{rcl} F(\xi\_3) &=& 32|E\_1(\xi\_3)| + 65|E\_2(\xi\_3)| + 116|E\_3(\xi\_3)| + 340|E\_4(\xi\_3)| + 149|E\_5(\xi\_3)| + 373|E\_6(\xi\_3)| + 147|E\_7(\xi\_3)|\\ &+ 200|E\_7(\xi\_3)| + 424|E\_8(\xi\_3)| + 648|E\_9(\xi\_3)|.\\ \end{array}$$

After some calculation, we have

$$\implies F(\emptyset\_3) = 19726 - 20096m + 6216m^2.$$

In the following theorem, we compute the Balaban index of the third type of rectangular hex-derived network, *ξ*3.

**Theorem 10.** *For the third type of rectangular hex-derived network ξ*3*, the Balaban index is equal to*

$$\begin{array}{rcl} J(\xi\_3) &=& \left(\frac{1}{315(15 - 28m + 14m^2)}\right) 7(-157 - 180\sqrt{2} + 12\sqrt{5} + 54\sqrt{10})m + 105(5 + 3\sqrt{2})m^2)(19 - 10m + 21m^2) \\ &+ 40m + 21m^2)(3(280 + 420\sqrt{2} - 70\sqrt{5} + 60\sqrt{7} - 231\sqrt{10} + 5\sqrt{14} + 6\sqrt{70})). \end{array}$$

**Proof.** Let *ξ*<sup>3</sup> be the rectangular hex-derived network *RHDN*3(*m*). By using edge partition from Table 5, the result follows. The Balaban index can be calculated by using (3) as follows.

$$J(\xi\_3) = \underbrace{\left(\frac{m}{m-n+2}\right)}\_{\text{init}\in\mathbb{E}(\xi\_3)} \frac{1}{\sqrt{\pi(\acute{m})\times\pi(\acute{n})}} = \left(\frac{m}{m-n+2}\right) \sum\_{\acute{m}\neq\mp\mathbb{E}\_{\overline{\mathbb{P}}}(\xi\_3)} \sum\_{j=1}^9 \frac{1}{\sqrt{\pi(\acute{m})\times\pi(\acute{n})}}$$

$$\begin{array}{rcl} J(\xi\_{3}) &=& \left(\frac{19 - 40m + 21m^{2}}{15 - 28m + 14m^{2}}\right) \left(\frac{1}{4}|E\_{1}(\xi\_{3})| + \frac{1}{2\sqrt{7}}|E\_{2}(\xi\_{3})| + \frac{1}{2\sqrt{10}}|E\_{3}(\xi\_{3})| + \frac{1}{6\sqrt{2}}|E\_{4}(\xi\_{3})| + \frac{1}{2}\right) \\ & \frac{1}{\sqrt{70}}|E\_{5}(\xi\_{3})| + \frac{1}{3\sqrt{14}}|E\_{6}(\xi\_{3})| + \frac{1}{10}|E\_{7}(\xi\_{3})| + \frac{1}{6\sqrt{5}}|E\_{8}(\xi\_{3})| + \frac{1}{18}|E\_{9}(\xi\_{3})|\end{array}$$

After some calculation, we have

$$\begin{split} l \Longrightarrow J(\vec{\xi}\_{3}) &= \ \left(\frac{1}{315(15 - 28m + 14m^{2})}\right) 7(-157 - 180\sqrt{2} + 12\sqrt{5} + 54\sqrt{10})m + 105(5 + 3\sqrt{2})m^{2}) \\ &\quad (19 - 40m + 21m^{2})(3(280 + 420\sqrt{2} - 70\sqrt{5} + 60\sqrt{7} - 231\sqrt{10} + 5\sqrt{14} + 6\sqrt{70})). \end{split}$$

Now, we compute *ReZG*1, *ReZG*<sup>2</sup> and *ReZG*<sup>3</sup> indices of the third type of rectangular hex-derived network *ξ*3.

**Theorem 11.** *Let ξ*<sup>3</sup> *be the third type of rectangular hex-derived network, then*

$$\bullet \qquad \mathrm{Re} Z G\_1(\xi\_3) = \frac{10102843}{32725} - \frac{2036 \mathrm{m}}{11} + \frac{861 \mathrm{m}^2}{11} \prime$$

$$\bullet \qquad \mathrm{Re}ZG\_2(\xi\_3) = 56 - 12m + 7m^2 \omega$$

• *ReZG*3(*ξ*3) *<sup>=</sup>* <sup>4</sup>(<sup>50785</sup> <sup>−</sup> <sup>50608</sup>*<sup>m</sup>* <sup>+</sup> <sup>13692</sup>*m*2)*.*

**Proof.** By using the edge partition given in Table 5, the ReZG1(*ξ*3) can be calculated by using (4) as follows.

$$\text{Re}ZG\_1(\underline{\mathfrak{r}}) \;= \sum\_{\substack{\mathfrak{m}\mathfrak{h}\in\mathbb{E}(\underline{\mathfrak{r}}\_3)}} \left( \frac{\mathfrak{r}(\mathfrak{m}) \times \mathfrak{r}(\mathfrak{n})}{\mathfrak{r}(\mathfrak{m}) + \mathfrak{r}(\mathfrak{n})} \right) = \sum\_{j=1}^9 \sum\_{\substack{\mathfrak{m}\mathfrak{h}\in E\_j(\underline{\mathfrak{r}}\_3)}} \left( \frac{\mathfrak{r}(\mathfrak{m}) \times \mathfrak{r}(\mathfrak{n})}{\mathfrak{r}(\mathfrak{m}) + \mathfrak{r}(\mathfrak{n})} \right).$$

$$\begin{split} \text{Re}\mathcal{Z}\mathcal{G}\_{1}(\underline{\mathcal{F}}\_{3}) &= 2|E\_{1}(\underline{\mathcal{F}}\_{3})| + \frac{28}{11}|E\_{2}(\underline{\mathcal{F}}\_{3})| + \frac{20}{7}|E\_{3}(\underline{\mathcal{F}}\_{3})| + \frac{36}{11}|E\_{4}(\underline{\mathcal{F}}\_{3})| + \frac{70}{17}|E\_{5}(\underline{\mathcal{F}}\_{3})| + \frac{126}{25}|E\_{6}(\underline{\mathcal{F}}\_{3})| + \frac{20}{7}|E\_{7}(\underline{\mathcal{F}}\_{3})| \\ &\leq |E\_{7}(\underline{\mathcal{F}}\_{3})| + \frac{45}{7}|E\_{8}(\underline{\mathcal{F}}\_{3})| + 9|E\_{9}(\underline{\mathcal{F}}\_{3})|. \end{split}$$

After some calculation, we have

$$\implies \text{Re}Z G\_1(\xi\_3) = \frac{10102843}{32725} - \frac{2036m}{11} + \frac{861m^2}{11}.$$

The ReZG2(*ξ*3) can be calculated by using (5) as follows.

$$\operatorname{ReZG}\_2(\mathbb{J}\_3) = \sum\_{\mathfrak{m}\mathfrak{n} \in \operatorname{E}(\mathbb{\tilde{E}}\_3)} \left( \frac{\mathfrak{r}(\mathfrak{m}) + \mathfrak{r}(\mathfrak{n})}{\mathfrak{r}(\mathfrak{m}) \times \mathfrak{r}(\mathfrak{n})} \right) = \sum\_{\mathfrak{m}\mathfrak{n} \in \operatorname{E}\_{\overline{\mathfrak{r}}}(\mathbb{\tilde{E}}\_3)} \sum\_{j=1}^9 \left( \frac{\mathfrak{r}(\mathfrak{m}) + \mathfrak{r}(\mathfrak{n})}{\mathfrak{r}(\mathfrak{m}) \times \mathfrak{r}(\mathfrak{n})} \right)$$

$$\begin{array}{rcl} \text{Re}\mathcal{Z}\mathcal{Z}\_{2}(\underline{\mathcal{F}}\_{3}) &=& \frac{1}{2}|E\_{1}(\underline{\mathcal{F}}\_{3})| + \frac{11}{28}|E\_{2}(\underline{\mathcal{F}}\_{3})| + \frac{7}{20}|E\_{3}(\underline{\mathcal{F}}\_{3})| + \frac{11}{36}|E\_{4}(\underline{\mathcal{F}}\_{3})| + \frac{17}{70}|E\_{5}(\underline{\mathcal{F}}\_{3})| + \frac{25}{126}|E\_{6}(\underline{\mathcal{F}}\_{3})| + \frac{1}{9}|E\_{6}(\underline{\mathcal{F}}\_{3})| \\ & \frac{1}{5}|E\_{7}(\underline{\mathcal{F}}\_{3})| + \frac{7}{45}|E\_{8}(\underline{\mathcal{F}}\_{3})| + \frac{1}{9}|E\_{9}(\underline{\mathcal{F}}\_{3})|. \end{array}$$

After some calculation, we have

$$\implies \text{Re}ZG\_2(\mathfrak{J}\_3) = 56 - 12m + 7m^2.$$

The ReZG3(*ξ*3) index can be calculated from (6) as follows.

$$\text{Re}Z G\_3(\mathfrak{f}\_3) \quad = \sum\_{\mathfrak{m} \not\equiv \mathbb{E}(\mathfrak{f}\_3)} (\mathfrak{r}(\mathfrak{m}) \times \mathfrak{r}(\mathfrak{n}))(\mathfrak{r}(\mathfrak{m}) + \mathfrak{r}(\mathfrak{n}) = \sum\_{\mathfrak{m} \not\equiv \mathbb{E}\_{\mathfrak{f}}(\mathfrak{f}\_3)} \sum\_{j=1}^{9} (\mathfrak{r}(\mathfrak{m}) \times \mathfrak{r}(\mathfrak{n}))(\mathfrak{r}(\mathfrak{m}) + \mathfrak{r}(\mathfrak{n})) $$

$$\begin{split} \text{Re}Z G\_3(\xi\_3) &= 128|E\_1(\xi\_3)| + 308|E\_2(\xi\_3)| + 560|E\_3(\xi\_3)| + 1584|E\_4(\xi\_3)| + 1190|E\_5(\xi\_3)| + 1190|E\_6(\xi\_3)| \\ &= 3150|E\_6(\xi\_3)| + 2000|E\_7(\xi\_3)| + 5040|E\_8(\xi\_3)| + 11664|E(\xi\_3)|. \end{split}$$

After some calculation, we have

$$\implies \text{R\z}ZG\_{3}(\mathfrak{f}\_{3})\; =\; \; 4(50785 - 50608m + 13692m^{2})\; .$$

Now, we compute *ABC*<sup>4</sup> and *GA*<sup>5</sup> indices of the third type of rectangular hex-derived network *ξ*3.

**Theorem 12.** *Let ξ*<sup>3</sup> *be the third type of rectangular hex-derived network, then*


**Proof.** By using the edge partition given in Table 6, the *ABC*4(*ξ*3) can be calculated by using (7) as follows.

*ABC*4(*ξ*3) = ∑*m*´ *n*´∈E(*ξ*3) *Sm*´ + *Sn*´ − 2 *Sm*´ *Sn*´ = ∑ *m*´ *n*´∈*Ej*(*ξ*3) ∑ *j*=10 *Sm*´ + *Sn*´ − 2 *Sm*´ *Sn*´ *ABC*4(*ξ*3) = <sup>1</sup> <sup>21</sup> <sup>|</sup>*E*10(*ξ*3)<sup>|</sup> <sup>+</sup> 6 <sup>|</sup>*E*11(*ξ*3)<sup>|</sup> <sup>+</sup> 83 <sup>|</sup>*E*12(*ξ*3)<sup>|</sup> <sup>+</sup> <sup>|</sup>*E*13(*ξ*3)<sup>|</sup> <sup>+</sup> √ |*E*14(*ξ*3)| + <sup>77</sup> <sup>|</sup>*E*15(*ξ*3)<sup>|</sup> <sup>+</sup> <sup>86</sup> <sup>|</sup>*E*16(*ξ*3)<sup>|</sup> <sup>+</sup> <sup>|</sup>*E*17(*ξ*3)<sup>|</sup> <sup>+</sup> √ |*E*18(*ξ*3)| + <sup>|</sup>*E*19(*ξ*3)<sup>|</sup> <sup>+</sup> <sup>|</sup>*E*20(*ξ*3)<sup>|</sup> <sup>+</sup> 10 <sup>|</sup>*E*21(*ξ*3)<sup>|</sup> <sup>+</sup> <sup>35</sup> <sup>|</sup>*E*22(*ξ*3)<sup>|</sup> <sup>+</sup> <sup>|</sup>*E*23(*ξ*3)<sup>|</sup> <sup>+</sup> <sup>97</sup> <sup>|</sup>*E*24(*ξ*3)<sup>|</sup> <sup>+</sup> <sup>|</sup>*E*25(*ξ*3)<sup>|</sup> <sup>+</sup> <sup>|</sup>*E*26(*ξ*3)<sup>|</sup> <sup>+</sup> 29 <sup>|</sup>*E*27(*ξ*3)<sup>|</sup> <sup>+</sup> <sup>43</sup> <sup>|</sup>*E*28(*ξ*3)<sup>|</sup> <sup>+</sup> 83 <sup>|</sup>*E*29(*ξ*3)<sup>|</sup> <sup>+</sup> 57 <sup>|</sup>*E*30(*ξ*3)<sup>|</sup> <sup>+</sup> 13 <sup>|</sup>*E*31(*ξ*3)<sup>|</sup> <sup>+</sup> 3 <sup>|</sup>*E*32(*ξ*3) + <sup>1</sup> <sup>|</sup>*E*33(*ξ*3) + <sup>|</sup>*E*34(*ξ*3)<sup>|</sup> <sup>+</sup> √ |*E*35(*ξ*3)| + <sup>|</sup>*E*36(*ξ*3)<sup>|</sup> <sup>+</sup> <sup>|</sup>*E*37(*ξ*3)<sup>|</sup> <sup>+</sup> <sup>67</sup> <sup>|</sup>*E*38(*ξ*3)<sup>|</sup> <sup>+</sup> 131 <sup>|</sup>*E*39(*ξ*3)<sup>|</sup> <sup>+</sup> 89 <sup>|</sup>*E*40(*ξ*3) + <sup>1</sup> <sup>283</sup> <sup>|</sup>*E*41(*ξ*3) + <sup>139</sup> <sup>|</sup>*E*42(*ξ*3)<sup>|</sup> <sup>+</sup> 7 <sup>|</sup>*E*43(*ξ*3)<sup>|</sup> <sup>+</sup> <sup>155</sup> <sup>|</sup>*E*44(*ξ*3)|.

After some calculation, we have

$$\begin{split} \text{sinc}\,AB\mathbb{C}\_{4}(\xi\_{5}) &= \ 22.459 + 8\sqrt{\frac{26}{55}}(-4+m) + 4\sqrt{\frac{58}{105}}(-4+m) + \frac{4}{7}\sqrt{\frac{67}{15}}(-4+m) + 3\sqrt{\frac{6}{13}} \\ & (-4+m)^{2} + 2\sqrt{\frac{26}{33}}(-3+m) + \frac{3}{11}\sqrt{\frac{43}{2}}(-3+m)^{2} + \sqrt{\frac{14}{65}}(-9+2m) + \frac{1}{35}\sqrt{\frac{139}{2}} \\ & (-9+2m) + \frac{1}{3}\sqrt{\frac{62}{7}}(-5+2m) + \frac{4}{63}\sqrt{31}(-5+2m) + \frac{4}{9}\sqrt{\frac{97}{7}}(-3+2m) + \frac{2}{21}\sqrt{89} \\ & (-3+2m) + \frac{1}{9}\sqrt{\frac{35}{2}}(-11+4m) + \frac{1}{78}\sqrt{\frac{155}{2}}(65-28m+3m^{2}). \end{split}$$

The *GA*5(*ξ*3) index can be calculated from (8) as follows.

$$GA\_5(\mathfrak{J}\_3) = \sum\_{\mathfrak{M}\mathfrak{h} \in \operatorname{E}(\mathfrak{J}\_3)} \frac{2\sqrt{S\_{\mathfrak{M}}S\_{\mathfrak{H}}}}{\left(S\_{\mathfrak{M}} + S\_{\mathfrak{H}}\right)} = \sum\_{\mathfrak{M}\mathfrak{h} \in E\_j(\mathfrak{J}\_3)} \sum\_{j=1}^{44} \frac{2\sqrt{S\_{\mathfrak{M}}S\_{\mathfrak{H}}}}{\left(S\_{\mathfrak{M}} + S\_{\mathfrak{H}}\right)^2}$$

*GA*5(*ξ*3) = 1|*E*10(*ξ*3)| + √ |*E*11(*ξ*3)| + √ |*E*12(*ξ*3)| + √ |*E*13(*ξ*3)| + <sup>|</sup>*E*14(*ξ*3)<sup>|</sup> <sup>+</sup> √ |*E*15(*ξ*3)| + √ |*E*16(*ξ*3)| + √ |*E*17(*ξ*3)| + <sup>|</sup>*E*18(*ξ*3)<sup>|</sup> <sup>+</sup> √ |*E*19(*ξ*3)| + √ |*E*20(*ξ*3)| + √ |*E*21(*ξ*3)| + 1|*E*22(*ξ*3)| + √ |*E*23(*ξ*3)| + √ |*E*24(*ξ*3)| + √ |*E*25(*ξ*3)| + √ |*E*26(*ξ*3)| + √ |*E*27(*ξ*3)| + |*E*28(*ξ*3)| + √ |*E*29(*ξ*3)| + √ |*E*30(*ξ*3)| + √ |*E*31(*ξ*3)| + √ |*E*32(*ξ*3)| + √ |*E*33(*ξ*3)| + √ |*E*34(*ξ*3)| + |*E*35(*ξ*3)| + √ |*E*36(*ξ*3)| + √ |*E*37(*ξ*3)| + √ |*E*38(*ξ*3)| + √ |*E*39(*ξ*3)| + √ |*E*40(*ξ*3)| + √ |*E*41(*ξ*3)| + 1|*E*42(*ξ*3)| + √ |*E*43(*ξ*3)| + 1|*E*44(*ξ*3)|.

After some calculations, we have

=⇒ *GA*5(*ξ*3) = 173.339 + √ (−<sup>4</sup> <sup>+</sup> *<sup>m</sup>*) + <sup>24</sup> √ (−<sup>4</sup> <sup>+</sup> *<sup>m</sup>*) + <sup>32</sup> √ (−<sup>4</sup> <sup>+</sup> *<sup>m</sup>*) + <sup>12</sup> √ (−<sup>4</sup> <sup>+</sup> *<sup>m</sup>*)<sup>2</sup> <sup>+</sup> √ (−<sup>3</sup> <sup>+</sup> *<sup>m</sup>*) <sup>−</sup> <sup>48</sup>*<sup>m</sup>* <sup>+</sup> <sup>9</sup>*m*<sup>2</sup> <sup>+</sup> √ (−<sup>9</sup> <sup>+</sup> <sup>2</sup>*m*) + <sup>3</sup> √ (−5 + 2*m*) + (−<sup>3</sup> <sup>+</sup> <sup>2</sup>*m*) + <sup>32</sup> √ (−3 + 2*m*).

The graphical representations of topological indices of these networks are depicted in Figures 5 and 6 for certain values of *m*. By varying the different values of *m*, the graphs are increasing. These graphs show the correctness of the results.

**Table 6.** Edge partition of the third type of rectangular hex-derived network *RHDN*3(*m*) based on the sum of degrees of end vertices of each edge.


**Figure 5.** Comparison of ABC4 index for *ξ*1, *ξ*<sup>2</sup> and *ξ*3.

**Figure 6.** Comparison of GA5 index for *ξ*1, *ξ*<sup>2</sup> and *ξ*3.

#### **3. Conclusions**

The study of topological descriptors are very useful to acquire the basic topologies of networks. In this paper, we find the exact results for Forgotten index, Balaban index, reclassified the Zagreb indices, ABC4 index and GA5 index of the Hex-derived networks of type 3. Due to their fascinating and challenging features, hex-derived networks have studied literature in relation to different graph-ideological parameters. However, their developmental circulatory features have been read for the foremost in this paper.

We are also very keen in designing some new networks and then study their topological indices which will be quite helpful to understand their primary priorities.

**Author Contributions:** Software, M.A.B.; validation, M.K.S. writing—original draft preparation, H.A.; writing—review and editing, W.G.; supervision, M.K.S.; funding acquisition, W.G.

**Funding:** This work has been partially supported by National Science Foundation of China (11761083).

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **References**


c 2019 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/).
