Lightweight design method and application of MEWP bracket based on multi-level optimization

Wen Li; Jian Wang; Zhanpeng Du; Hongfeng Ma; Lijie Zhang; Libin Duan; Wen Li; Jian Wang; Zhanpeng Du; Hongfeng Ma; Lijie Zhang; Libin Duan

doi:10.3934/era.2022224

Electronic Research Archive

2022, Volume 30, Issue 12: 4416-4435. doi: 10.3934/era.2022224

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Research article Special Issues

Lightweight design method and application of MEWP bracket based on multi-level optimization

1.
Hebei Provincial Key Laboratory of Heavy Machinery Fluid Power Transmission and Control, Yanshan University, Qinhuangdao 066004, China
2.
Jiangsu XCMG State Key Laboratory Technology Co., Ltd., Xuzhou 221000, China
3.
Institute of Lightweight and Safety of New Energy Vehicle, School of Automotive and Traffic Engineering, Jiangsu University, Zhenjiang 212013, China

Received: 13 August 2022 Revised: 29 September 2022 Accepted: 30 September 2022 Published: 13 October 2022

Mobile elevating work platform (MEWP) is a large-scale engineering machinery equipment that transports workers and tools to the designated height for operation. As the key supporting component of MEWP, the bracket simultaneously needs to meet the performances of high stiffness and strength. Furthermore, the mechanical performance of the bracket can be significantly influenced by its cross-sectional shape. However, the optimal cross-sectional shape of bracket is not easily to obtain owing to the lacking of lightweight design method. Thus, a lightweight design method of MEWP bracket based on multi-level optimization is proposed in this paper. Firstly, the multi-case topology optimization model of MEWP bracket is constructed by using the compromise programming method, and the optimal section configuration of MEWP bracket is obtained based on Solid Isotropic Material with Penalization (SIMP). Secondly, the parameterization of the cross-sectional shape of bracket is realized using the mesh deformation technology, and the multi-case optimization mathematical model of the MEWP bracket is established. Then, the cross-sectional shape and gauge of the bracket are optimized using multi-level optimization method. The optimized results show that weight reduction mass is 11.66 kg and the ratio is 52.4% under the premise that the stiffness of the bracket does not decrease. Furthermore, the weight of MEWP bracket optimized by the multi-level optimization method reduced by 1.27 kg compared with single gauge optimization method. Finally, a physical prototype is developed according to the optimization results.

Keywords:

Citation: Wen Li, Jian Wang, Zhanpeng Du, Hongfeng Ma, Lijie Zhang, Libin Duan. Lightweight design method and application of MEWP bracket based on multi-level optimization[J]. Electronic Research Archive, 2022, 30(12): 4416-4435. doi: 10.3934/era.2022224

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Abstract

1. Introduction

Graph theory has provided the researcher with various useful tools, such as graph labeling, locating numbers and topological indices. Graph theory subject has many applications and implementations in different research subjects like chemistry, medicine and engineering. A graph can be recognized by a numeric value, a polynomial, a sequence of numbers or a matrix. The representation of the chemical compound in terms of diagram, known as its molecular graph, in which its atoms and the chemical bonding between them represent the nodes and edges, respectively. Recently, a new subject caught attention of the researchers was introduced, which is the combination of chemistry, information science and mathematics is called Cheminformatics, which studies QSAR/QSPR relationship, bioactivity and characterization of chemical compounds ^[1].

The topological index is a numeric value related with chemical compositions maintaining the correlation of chemical structures with many physico-chemical properties, chemical reactivity or biological activity. Topological indices are prepared on the grounds of the transformation of a chemical network into a number that describes the topology of the chemical network. Some of the main types of topological indices of graphs are distance-based topological indices, degree-based topological indices, and counting-related topological indices. Recently, numerous researchers have found topological indices for the study of fundamental properties of molecular graph or network. These networks have very motivating topological properties which have been considered in different characteristics in ^{[2,3,4,5,6,7,8,9]}.

Let $G = (V, E)$ be a simple connected graph, with $V$ be the vertex set and $E$ be the edge set of graph $G,$ with order $|V| = p$ , size $|E| = q.$ The number of edges incident with a vertex $\omega$ is known as the degree of $\omega$ , denoted by $\zeta(\omega)$ . The reverse vertex degree ( $\mathfrak{R}(\omega)$ ) was introduced by Kulli ^[10] defined as: $\mathfrak{R}(\omega) = 1-\zeta(\omega)+\Delta,$ where $\Delta$ denoted the maximum degree of the given graph. Let $\mathbf{E}_{\mathfrak{R}(\omega), \mathfrak{R}(\mu)}$ represents the edge partition of the given graph based on reverse degree of end vertices of an edge $\omega\mu\in E$ and $|\mathbf{E}_{\mathfrak{R}(\omega), \mathfrak{R}(\mu)}|$ represents its cardinality. There are detailed variations of topological indices mainly distance-based and degree-based indices, see ^{[11,12,13,14,15,16]}. Milan Randic ^[30] was the first who defined the degree-based indices and its reverse Randic index is defined as:

$\begin{equation} \mathfrak{R}\mathbb{R}_{\alpha}(G) = \sum\limits_{\omega\mu\in E(G)}\bigg( \mathfrak{R}(\omega)\times \mathfrak{R}(\mu) \bigg)^{\alpha}, \, \, \, \alpha = \frac{1}{2}, -\frac{1}{2}, 1, -1. \end{equation}$

(1.1)

Estrada et al. presented the atom bond connectivity ( $\mathbb{ABC}$ ) index in ^[18] and the reverse atom bond connectivity ( $\mathfrak{R}\mathbb{ABC}$ ) is defined as:

$\begin{equation} \mathfrak{R}\mathbb{ABC}(G) = \sum\limits_{\omega\mu\in E(G)}\sqrt{\frac{ \mathfrak{R}(\omega)+ \mathfrak{R}(\mu)-2}{ \mathfrak{R}(\omega)\times \mathfrak{R}(\mu)}} \end{equation}$

(1.2)

Vukicevic and Furtula defined the geometric arithmetic ( $\mathbb{GA}$ ) index in ^[19] and the reverse geometric arithmetic ( $\mathfrak{R}\mathbb{GA}$ ) is presented as:

$\begin{equation} \mathfrak{R}\mathbb{GA}(G) = \sum\limits_{\omega\mu\in E(G)}\frac{ 2\, \sqrt{\mathfrak{R}(\omega)\times \mathfrak{R}(\mu)}}{ \mathfrak{R}(\omega)+ \mathfrak{R}(\mu)} \end{equation}$

(1.3)

Gutman et al. ^[20,21] defined the first and second Zagreb and its reverse indices as:

$\begin{eqnarray} \mathfrak{R}\mathbb{M}_1(G) & = & \sum\limits_{\omega\mu\in E(G)}\bigg(\mathfrak{R}(\omega)+ \mathfrak{R}(\mu)\bigg) \end{eqnarray}$

(1.4)

$\begin{eqnarray} \mathfrak{R}\mathbb{M}_2(G) & = & \sum\limits_{\omega\mu\in E(G)}\bigg(\mathfrak{R}(\omega)\times \mathfrak{R}(\mu)\bigg) \end{eqnarray}$

(1.5)

Shirdel et al. ^[22] introduced hyper Zagreb index. We defined the reverse hyper Zagreb index as:

$\begin{equation} \mathfrak{R}\mathbb{HM}(G) = \sum\limits_{\omega\mu\in E(G)}\bigg( \mathfrak{R}(\omega)+ \mathfrak{R}(\mu) \bigg)^{2} \end{equation}$

(1.6)

Furtula and Gutman ^[23] accomplished the forgotten index and its reverse forgotten index as:

$\begin{equation} \mathfrak{R}\mathbb{F}(G) = \sum\limits_{\omega\mu\in E(G)}\bigg( \left(\mathfrak{R}(\omega)\right)^2+ \left(\mathfrak{R}(\mu)\right)^{2} \bigg) \end{equation}$

(1.7)

Augmented Zagreb index was introduced by Furtula et al. ^[24] and the reverse augmented Zagreb index as:

$\begin{equation} \mathfrak{R}\mathbb{AZI}(G) = \sum\limits_{\omega\mu\in E(G)}\bigg( \frac{\mathfrak{R}(\omega)\times \mathfrak{R}(\mu)}{\mathfrak{R}(\omega)+ \mathfrak{R}(\mu)-2} \bigg)^{3} \end{equation}$

(1.8)

Ranjini et al. ^[25] introduced the first redefined, second redefined and third redefined Zagreb indices. The reverse first redefined, second redefined and third redefined Zagreb indices are defined as:

$\begin{eqnarray} \mathfrak{R}\mathbb{ReZ}_1(G) & = & \sum\limits_{\omega\mu\in E(G)} \frac{\mathfrak{R}(\omega)+ \mathfrak{R}(\mu)}{\mathfrak{R}(\omega)\times \mathfrak{R}(\mu)} \end{eqnarray}$

(1.9)

$\begin{eqnarray} \mathfrak{R}\mathbb{ReZ}_2(G) & = & \sum\limits_{\omega\mu\in E(G)} \frac{\mathfrak{R}(\omega)\times \mathfrak{R}(\mu)}{\mathfrak{R}(\omega)+ \mathfrak{R}(\mu)} \end{eqnarray}$

(1.10)

$\begin{eqnarray} \mathfrak{R}\mathbb{ReZ}_3(G) & = & \sum\limits_{\omega\mu\in E(G)} \bigg(\mathfrak{R}(\omega)+ \mathfrak{R}(\mu)\bigg)\bigg(\mathfrak{R}(\omega)\times \mathfrak{R}(\mu)\bigg) \end{eqnarray}$

(1.11)

For latest results on topological indices see ^{[26,27,28,29,31,32,33,34,35]}. In this paper, we compute the exact results for all the above reverse indices.

2. Structure of third type hex-derived networks

With the help of complete graphs of order 3 ( $K_3$ ), Chen et al. ^[36] assembled a hexagonal mesh. In terms of chemistry, these $K_3$ graphs are also called oxide graphs. The is obtained by joining these $K_3$ graphs. Two dimensional mesh graph HX(2) (see ), is obtained by joining six $K_3$ graphs and three dimensional mesh graph HX(3) (see ) is obtained by putting $K_3$ graphs around all side of HX(2). Furthermore, repeating the same process by putting the $t$ $K_3$ graph around each hexagon, we obtained the $t^{th}$ hexagonal mesh. To be noted that the one dimensional hexagonal mesh graph does not exist.

Figure 1. Hexagonal meshes: (a)

$HX(2)$ and (b)

$HX(3)$ .

DownLoad: Full-Size Img PowerPoint

Simonraj et al. ^[37] created the new network which is named as third type of hex-derived networks. The graphically construction algorithm for third type of hexagonal hex-derived network $HHDN_3(t)$ (see ), triangular hex-derived network $THDN_3(t)$ (see ) and rectangular hex-derived network $RHDN_3(t)$ (see Figure 4) are defined in ^[38,39] and they determined some topological indices of these new derived networks. Some networks such as hexagonal, honeycomb, and grid networks, for instance, endure closeness to atomic or molecular lattice configurations. Related research that applies this theory and which could get additional advantages from the visions of the new research is found in ^{[40,41,42,43,44,45,46]}.

Figure 2. Third type of hexagonal hex-derived network

$HHDN_3(t)$ for

$t = 4$ .

DownLoad: Full-Size Img PowerPoint

Figure 3. Third type of triangular hex-derived network

$THDN_3(t)$ for

$t = 7$ .

DownLoad: Full-Size Img PowerPoint

Figure 4. Third type of rectangular hex-derived network

$RHDN_3(t)$ for

$t = 4$ .

DownLoad: Full-Size Img PowerPoint

3. Hexagonal hex-derived network $HHDN_3(t)$

Let $\Gamma_1 = HHDN_3(t)$ be the third type of hexagonal hex-derived network which is shown in , where $t \ge 4.$ The graph $\Gamma_1$ has $21t^2 -39t + 19$ vertices from which $18t^2-36t+18$ vertices of reverse degree $15$ , $4$ vertices of reverse degree $12$ , $6t-12$ vertices of reverse degree $9$ and $3t^2-9t+9$ vertices of reverse degree $1.$ There are $63t^2-123t+60$ number of edges of $\Gamma_1$ is partitioned into nine classes based on their reverse degrees which are given in Eq (3.1).

$\begin{equation} |\mathbf{E}_{\mathfrak{R}(\omega), \mathfrak{R}(\mu)}(\Gamma_1)| = \left\{ \begin{array}{ll} 9t^2-33t+30, & \hbox{for;}\ \mathfrak{R}(\omega) = 1, \mathfrak{R}(\mu) = 1 \\ 12t-24, & \hbox{for;}\ \mathfrak{R}(\omega) = 9, \mathfrak{R}(\mu) = 1 \\ 6t-18, & \hbox{for;}\ \mathfrak{R}(\omega) = 9, \mathfrak{R}(\mu) = 9 \\ 6, & \hbox{for;}\ \mathfrak{R}(\omega) = 12, \mathfrak{R}(\mu) = 1 \\ 12, & \hbox{for;}\ \mathfrak{R}(\omega) = 12, \mathfrak{R}(\mu) = 9 \\ 36t^2-108t+84, & \hbox{for;}\ \mathfrak{R}(\omega) = 15, \mathfrak{R}(\mu) = 1 \\ 36t-72, & \hbox{for;}\ \mathfrak{R}(\omega) = 15, \mathfrak{R}(\mu) = 9 \\ 24, & \hbox{for;}\ \mathfrak{R}(\omega) = 15, \mathfrak{R}(\mu) = 12 \\ 18t^2-36t+18, & \hbox{for;}\ \mathfrak{R}(\omega) = 15, \mathfrak{R}(\mu) = 15 \end{array} \right. \end{equation}$

(3.1)

In the next theorem, we will calculate the reverse Randic index, reverse Zagreb indices, reverse forgotten index for $\Gamma_1$ graph.

Theorem 3.1. Let $\Gamma_1$ be the third type of hexagonal hex-derived network, then

● $\mathfrak{R}\mathbb{R}_{\alpha}(\Gamma_1) = [9+36(15)^{\alpha}+18(225)^{\alpha}]t^2+[-33+12(9)^{\alpha}+6(81)^{\alpha}-108(15)^{\alpha}+36(135)^{\alpha}-36(225)^{\alpha}]t +30-24(9)^{\alpha}-18(81)^{\alpha}+6(12)^{\alpha}+12(108)^{\alpha}+84(15)^{\alpha}-72(135)^{\alpha}+24(180)^{\alpha}+18(225)^{\alpha}$

● $\mathfrak{R}\mathbb{M}_1(\Gamma_1) = 1134\, {t}^{2}-1782\, t+630$

● $\mathfrak{R}\mathbb{M}_2(\Gamma_1) = 4599\, {t}^{2}-4299\, t-366$

● $\mathfrak{R}\mathbb{HM}(\Gamma_1) = 25452\, {t}^{2}-36300\, t+11922$

● $\mathfrak{R}\mathbb{F}(\Gamma_1) = 16254\, {t}^{2}-27702\, t+12654$

Proof. Let $\Gamma_1$ be the third type of hexagonal hex-derived network which is shown in . The order of hexagonal hex derived network $\Gamma_1$ is $p = |\Gamma_1| = 21t^2 -39t + 19$ and size is $q = 63t^2-123t+60.$ The edge partitioned of $\Gamma_1$ based on their reverse degrees are shown in Eq (3.1). Reverse Randic index can be calculated by using Eq (3.1). Thus, from Eq (1.1), it follows,

$\mathfrak{R}\mathbb{R}_{\alpha}(\Gamma_1) = \left.(1)^{\alpha}|\mathbf{E}_{1, 1}(\Gamma_1)|+(9)^{\alpha}|\mathbf{E}_{9, 1}(\Gamma_1)| +(81)^{\alpha}|\mathbf{E}_{9, 9}(\Gamma_1)|+(12)^{\alpha}|\mathbf{E}_{12, 1}(\Gamma_1)|\right.\\ +(108)^{\alpha}|\mathbf{E}_{12, 9}(\Gamma_1)| +(15)^{\alpha}|\mathbf{E}_{15, 1}(\Gamma_1)| +(135)^{\alpha}|\mathbf{E}_{15, 9}(\Gamma_1)|+(180)^{\alpha}|\mathbf{E}_{15, 12}(\Gamma_1)| +(225)^{\alpha}|\mathbf{E}_{15, 15}(\Gamma_1)|.$

$= (9t^2-33t+30)+(9)^{\alpha}(12t-24) +(81)^{\alpha}(6t-18)+(12)^{\alpha}(6) +(108)^{\alpha}(12) +(15)^{\alpha}(36t^2-108t+84) +(135)^{\alpha}(36t-72)+(180)^{\alpha}(24) +(225)^{\alpha}(18t^2-36t+18).$

$= [9+36(15)^{\alpha}+18(225)^{\alpha}]t^2+[-33+12(9)^{\alpha}+6(81)^{\alpha}-108(15)^{\alpha}+36(135)^{\alpha}-36(225)^{\alpha}]t +30-24(9)^{\alpha}-18(81)^{\alpha}+6(12)^{\alpha}+12(108)^{\alpha}+84(15)^{\alpha}-72(135)^{\alpha}+24(180)^{\alpha}+18(225)^{\alpha}.$ Put $\alpha = 1$ and after some calculation, we get reverse second Zagreb index $\mathfrak{R}\mathbb{M}_2(\Gamma_1)$ as:

$\begin{equation} \mathfrak{R}\mathbb{M}_2(\Gamma_1) = 4599\, {t}^{2}-4299\, t-366. \end{equation}$

(3.2)

Using the Eq (1.4), we can determine the reverse first Zagreb index $\mathfrak{R}\mathbb{M}_1(\Gamma_1)$ as:

$\mathfrak{R}\mathbb{M}_1(\Gamma_1) = 2\times|\mathbf{E}_{1, 1}(\Gamma_1)|+10\times|\mathbf{E}_{9, 1}(\Gamma_1)| +18\times|\mathbf{E}_{9, 9}(\Gamma_1)|+13\times|\mathbf{E}_{12, 1}(\Gamma_1)|+21\times|\mathbf{E}_{12, 9}(\Gamma_1)| +16\times|\mathbf{E}_{15, 1}(\Gamma_1)| +24\times|\mathbf{E}_{15, 9}(\Gamma_1)|+27\times|\mathbf{E}_{15, 12}(\Gamma_1)| +30\times|\mathbf{E}_{15, 15}(\Gamma_1)|.$

By putting the values of from equation (3.1) and after simplification, we obtain:

$\begin{equation} \mathfrak{R}\mathbb{M}_1(\Gamma_1) = 1134\, {t}^{2}-1782\, t+630. \end{equation}$

(3.3)

Using the Eq (1.6), we can determine the reverse hyper Zagreb index $\mathfrak{R}\mathbb{HM}(\Gamma_1)$ as:

$\mathfrak{R}\mathbb{HM}(\Gamma_1) = 4\times|\mathbf{E}_{1, 1}(\Gamma_1)|+100\times|\mathbf{E}_{9, 1}(\Gamma_1)| +324\times|\mathbf{E}_{9, 9}(\Gamma_1)|+169\times|\mathbf{E}_{12, 1}(\Gamma_1)|+441\times|\mathbf{E}_{12, 9}(\Gamma_1)| +256\times|\mathbf{E}_{15, 1}(\Gamma_1)| +576\times|\mathbf{E}_{15, 9}(\Gamma_1)|+729\times|\mathbf{E}_{15, 12}(\Gamma_1)| +900\times|\mathbf{E}_{15, 15}(\Gamma_1)|.$

After simplification, we get

$\begin{equation*} \mathfrak{R}\mathbb{HM}(\Gamma_1) = 25452\, {t}^{2}-36300\, t+11922 \end{equation*}$

Using the Eq (1.7), we can determine the reverse forgotten index $\mathfrak{R}\mathbb{F}(\Gamma_1))$ as:

$\mathfrak{R}\mathbb{F}(\Gamma_1)) = 2\times|\mathbf{E}_{1, 1}(\Gamma_1)|+82\times|\mathbf{E}_{9, 1}(\Gamma_1)| +162\times|\mathbf{E}_{9, 9}(\Gamma_1)|+145\times|\mathbf{E}_{12, 1}(\Gamma_1)|+225\times|\mathbf{E}_{12, 9}(\Gamma_1)| +226\times|\mathbf{E}_{15, 1}(\Gamma_1)| +306\times|\mathbf{E}_{15, 9}(\Gamma_1)|+369\times|\mathbf{E}_{15, 12}(\Gamma_1)| +450\times|\mathbf{E}_{15, 15}(\Gamma_1)|.$

After simplification, we get

$\begin{equation*} \mathfrak{R}\mathbb{F}(\Gamma_1) = 16254\, {t}^{2}-27702\, t+12654 \end{equation*}$

In the next theorem, we will calculate the reverse atom bond connectivity index, reverse geometric arithmetic index for $\Gamma_1$ graph.

Theorem 3.2. Let $\Gamma_1$ be the third type of hexagonal hex-derived network, then

● $\mathfrak{R}\mathbb{ABC}(\Gamma_1) = \left({\frac {12\, \sqrt {210}}{5}}+{\frac {12\, \sqrt {7}}{5}} \right) {t}^{2}+ \left(8\, \sqrt {2}+\frac{8}{3}-{\frac {36\, \sqrt {210}}{5}} +\frac{4\, \sqrt {330}}{5}-{\frac {24\, \sqrt {7}}{5}} \right) t-8-16\, \sqrt {2} +\sqrt {33}+\frac{2\, \sqrt {57}}{3}+{\frac {28\, \sqrt {210}}{5}}-\frac{8\, \sqrt { 330}}{5}+4\, \sqrt {5}+{\frac {12\, \sqrt {7}}{5}}$

● $\mathfrak{R}\mathbb{GA}(\Gamma_1) = \left(27+\frac{9\, \sqrt {15}}{2} \right) {t}^{2}+ \left(-{\frac{279}{5}}-\frac{9\, \sqrt {15}}{2} \right) t+{\frac{78}{5}}+{\frac {792\, \sqrt {3}}{91}}-\frac{15\, \sqrt {15}}{2}+{\frac {32\, \sqrt {5}}{3}}.$

Proof. The reverse atom bond connectivity ( $\mathfrak{R}\mathbb{ABC}(\Gamma_1)$ ), can be determined by using Eq (1.2) and Eq (3.1), as follows:

$\mathfrak{R}\mathbb{ABC}(\Gamma_1) = 0\times|\mathbf{E}_{1, 1}(\Gamma_1)|+\sqrt{\frac{8}{9}}\times|\mathbf{E}_{9, 1}(\Gamma_1)| +\sqrt{\frac{16}{81}}\times|\mathbf{E}_{9, 9}(\Gamma_1)|+\sqrt{\frac{11}{12}}\times|\mathbf{E}_{12, 1}(\Gamma_1)| +\sqrt{\frac{19}{108}}\times|\mathbf{E}_{12, 9}(\Gamma_1)| +\sqrt{\frac{14}{15}}\times|\mathbf{E}_{15, 1}(\Gamma_1)| +\sqrt{\frac{22}{135}}\times|\mathbf{E}_{15, 9}(\Gamma_1)|+\sqrt{\frac{25}{180}}\times|\mathbf{E}_{15, 12}(\Gamma_1)| +\sqrt{\frac{28}{225}}\times|\mathbf{E}_{15, 15}(\Gamma_1)|.$

After some simplification, we get

$\mathfrak{R}\mathbb{ABC}(\Gamma_1) = \left({\frac {12\, \sqrt {210}}{5}}+{\frac {12\, \sqrt {7}}{5}} \right) {t}^{2}+ \left(8\, \sqrt {2}+\frac{8}{3}-{\frac {36\, \sqrt {210}}{5}} +\frac{4\, \sqrt {330}}{5}-{\frac {24\, \sqrt {7}}{5}} \right) t-8-16\, \sqrt {2} +\sqrt {33}+\frac{2\, \sqrt {57}}{3}+{\frac {28\, \sqrt {210}}{5}}-\frac{8\, \sqrt { 330}}{5}+4\, \sqrt {5}+{\frac {12\, \sqrt {7}}{5}}.$

The reverse geometric arithmetic ( $\mathfrak{R}\mathbb{GA}(\Gamma_1)$ ), can be determined by using Eq (1.3) and Eq (3.1), as follows:

$\mathfrak{R}\mathbb{GA}(\Gamma_1) = |\mathbf{E}_{1, 1}(\Gamma_1)|+\frac{2\, \sqrt{9}}{10}\times|\mathbf{E}_{9, 1}(\Gamma_1)| +\frac{2\, \sqrt{81}}{18}\times|\mathbf{E}_{9, 9}(\Gamma_1)|+\frac{2\, \sqrt{12}}{13}\times|\mathbf{E}_{12, 1}(\Gamma_1)| +\frac{2\, \sqrt{108}}{21}\times|\mathbf{E}_{12, 9}(\Gamma_1)| +\frac{2\, \sqrt{15}}{16}\times|\mathbf{E}_{15, 1}(\Gamma_1)| +\frac{2\, \sqrt{135}}{24}\times|\mathbf{E}_{15, 9}(\Gamma_1)|+\frac{2\, \sqrt{180}}{27}\times|\mathbf{E}_{15, 12}(\Gamma_1)| +\frac{2\, \sqrt{225}}{30}\times|\mathbf{E}_{15, 15}(\Gamma_1)|.$

After some simplification, we get

$\mathfrak{R}\mathbb{GA}(\Gamma_1) = \left(27+\frac{9\, \sqrt {15}}{2} \right) {t}^{2}+ \left(-{\frac{279}{5}}-\frac{9\, \sqrt {15}}{2} \right) t+{\frac{78}{5}}+{\frac {792\, \sqrt {3}}{91}}-\frac{15\, \sqrt {15}}{2}+{\frac {32\, \sqrt {5}}{3}}.$ In the next theorems, we will calculate the reverse redefined Zagreb indices for $\Gamma_1$ graph.

Theorem 3.3. Let $\Gamma_1$ be the third type of hexagonal hex-derived network, then

● $\mathfrak{R}\mathbb{ReZ}_1(\Gamma_1) = {\frac {294\, {t}^{2}}{5}}-{\frac {2474\, t}{15}}+{\frac{3629}{30}}$

● $\mathfrak{R}\mathbb{ReZ}_2(\Gamma_1) = {\frac {693\, {t}^{2}}{4}}-{\frac {2949\, t}{20}}-{\frac{93907}{1820}}$

● $\mathfrak{R}\mathbb{ReZ}_3(\Gamma_1) = 130158\, {t}^{2}-142518\, t+24828$

Proof. Reverse redefined Zagreb indices can be calculated by using Eq (3.1), the $\mathfrak{R}\mathbb{ReZ}_1(\Gamma_1)$ by using Eq (1.9) as follows:

$\mathfrak{R}\mathbb{ReZ}_1(\Gamma_1) = 2\times|\mathbf{E}_{1, 1}(\Gamma_1)|+\frac{10}{9}\times|\mathbf{E}_{9, 1}(\Gamma_1)| +\frac{18}{81}\times|\mathbf{E}_{9, 9}(\Gamma_1)|+\frac{13}{12}\times|\mathbf{E}_{12, 1}(\Gamma_1)| +\frac{21}{108}\times|\mathbf{E}_{12, 9}(\Gamma_1)| +\frac{16}{15}\times|\mathbf{E}_{15, 1}(\Gamma_1)| +\frac{24}{135}\times|\mathbf{E}_{15, 9}(\Gamma_1)|+\frac{27}{108}\times|\mathbf{E}_{15, 12}(\Gamma_1)| +\frac{30}{225}\times|\mathbf{E}_{15, 15}(\Gamma_1)|.$

After some simplification, we get

$\mathfrak{R}\mathbb{ReZ}_1(\Gamma_1) = {\frac {294\, {t}^{2}}{5}}-{\frac {2474\, t}{15}}+{\frac{3629}{30}}.$

The $\mathfrak{R}\mathbb{ReZ}_2(\Gamma_1)$ can be determined by using Eq (1.10) as follows:

$\mathfrak{R}\mathbb{ReZ}_2(\Gamma_1) = \frac{1}{2}\times|\mathbf{E}_{1, 1}(\Gamma_1)|+\frac{9}{10}\times|\mathbf{E}_{9, 1}(\Gamma_1)| +\frac{81}{18}\times|\mathbf{E}_{9, 9}(\Gamma_1)|+\frac{12}{13}\times|\mathbf{E}_{12, 1}(\Gamma_1)| +\frac{108}{21}\times|\mathbf{E}_{12, 9}(\Gamma_1)| +\frac{15}{16}\times|\mathbf{E}_{15, 1}(\Gamma_1)| +\frac{135}{24}\times|\mathbf{E}_{15, 9}(\Gamma_1)|+\frac{108}{27}\times|\mathbf{E}_{15, 12}(\Gamma_1)| +\frac{225}{30}\times|\mathbf{E}_{15, 15}(\Gamma_1)|.$

After some simplification, we get

$\mathfrak{R}\mathbb{ReZ}_2(\Gamma_1) = {\frac {693\, {t}^{2}}{4}}-{\frac {2949\, t}{20}}-{\frac{93907}{1820}}.$

The $\mathfrak{R}\mathbb{ReZ}_3(\Gamma_1)$ can be calculated by using Eq (1.11) as follows:

$\mathfrak{R}\mathbb{ReZ}_3(\Gamma_1) = 2\times|\mathbf{E}_{1, 1}(\Gamma_1)|+90\times|\mathbf{E}_{9, 1}(\Gamma_1)| +1458\times|\mathbf{E}_{9, 9}(\Gamma_1)|+156\times|\mathbf{E}_{12, 1}(\Gamma_1)| +2268\times|\mathbf{E}_{12, 9}(\Gamma_1)| +240\times|\mathbf{E}_{15, 1}(\Gamma_1)| +3240\times|\mathbf{E}_{15, 9}(\Gamma_1)|+2916\times|\mathbf{E}_{15, 12}(\Gamma_1)| +6750\times|\mathbf{E}_{15, 15}(\Gamma_1)|.$

After some simplification, we get

$\mathfrak{R}\mathbb{ReZ}_3(\Gamma_1) = 130158\, {t}^{2}-142518\, t+24828.$

4. Triangular hex-derived network $THDN3(p)$

Let $\Gamma_2 = THDN_3(t)$ be the third type of triangular hex-derived network which is shown in , where $t \ge 4.$ The graph $\Gamma_2$ has $\frac{7t^2-11t+6}{2}$ vertices. There are $\frac{21t^2-39t+18}{2}$ number of edges of $\Gamma_2$ is partitioned into six classes based on their reverse degrees which are given in Eq (4.1). Now we calculated reverse degree based indices such as: reverse Randic index $\mathfrak{R}\mathbb{R}_{\alpha},$ reverse atom bond connectivity index $\mathfrak{R}\mathbb{ABC},$ reverse geometric arithmetic index $\mathfrak{R}\mathbb{GA},$ first reverse Zagreb index $\mathfrak{R}\mathbb{M}_1,$ second reverse Zagreb index $\mathfrak{R}\mathbb{M}_2$ , reverse forgotten index $\mathfrak{R}\mathbb{F},$ reverse hyper Zagreb index $\mathfrak{R}\mathbb{HM}$ and reverse redefined Zagreb indices for $\Gamma_2$ graph.

$\begin{equation} |\mathbf{E}_{\mathfrak{R}(\omega), \mathfrak{R}(\mu)}(\Gamma_2)| = \left\{ \begin{array}{ll} \frac{3t^2}{2}-\frac{21t}{2}+18, & \hbox{for;}\ \mathfrak{R}(\omega) = 1, \mathfrak{R}(\mu) = 1 \\ 6t-18, & \hbox{for;}\ \mathfrak{R}(\omega) = 9, \mathfrak{R}(\mu) = 1 \\ 3t-6, & \hbox{for;}\ \mathfrak{R}(\omega) = 9, \mathfrak{R}(\mu) = 9 \\ 6t^2-30t+36, & \hbox{for;}\ \mathfrak{R}(\omega) = 15, \mathfrak{R}(\mu) = 1 \\ 18t-30, & \hbox{for;}\ \mathfrak{R}(\omega) = 15, \mathfrak{R}(\mu) = 9 \\ 3t^2-6t+9, & \hbox{for;}\ \mathfrak{R}(\omega) = 15, \mathfrak{R}(\mu) = 15 \end{array} \right. \end{equation}$

(4.1)

In the next theorem, we will calculate the reverse Randic index, reverse Zagreb indices, reverse forgotten index for $\Gamma_2$ graph.

Theorem 4.1. Let $\Gamma_2$ be the third type of triangular hex-derived network, then

● $\mathfrak{R}\mathbb{R}_{\alpha}(\Gamma_2) = \left(\frac{3}{2}+6\, (15)^{\alpha}+3\, (225)^{\alpha} \right) {t}^{2}+ \left(\frac{-21}{2}+6\, (9)^{\alpha}+3\, (81)^{\alpha}-30\, (15)^{\alpha}+\right. \\ \left. 18\, ( 135)^{\alpha}-6\, (225)^{\alpha} \right) t+18-18\, (9)^{\alpha}-6\, (81)^ {\alpha}+36\, (15)^{\alpha}-30\, (135)^{\alpha}+9\, (225)^{\alpha}.$

● $\mathfrak{R}\mathbb{M}_1(\Gamma_2) = 189\, {t}^{2}-135\, t-126$

● $\mathfrak{R}\mathbb{M}_2(\Gamma_2) = {\frac {1533\, {t}^{2}}{2}}+{\frac {1833\, t}{2}}-2115$

● $\mathfrak{R}\mathbb{HM}(\Gamma_2) = 4242\, {t}^{2}-1182\, t-3636$

● $\mathfrak{R}\mathbb{F}(\Gamma_2) = 2709\, {t}^{2}-3015\, t+594$

Proof. Let $\Gamma_2$ be the third type of triangular hex-derived network which is shown in . The order of triangular hex derived network $\Gamma_2$ is $p = |\Gamma_2| = \frac{7t^2-11t+6}{2}$ and size is $q = \frac{21t^2-39t+18}{2}.$ The edge partitioned of $\Gamma_2$ based on their reverse degrees are shown in Eq (4.1). Reverse Randic index can be calculated by using Eq (4.1). Thus, from Eq (1.1), it follows,

$\mathfrak{R}\mathbb{R}_{\alpha}(\Gamma_2) = (1)^{\alpha}|\mathbf{E}_{1, 1}(\Gamma_2)|+(9)^{\alpha}|\mathbf{E}_{9, 1}(\Gamma_2)| +(81)^{\alpha}|\mathbf{E}_{9, 9}(\Gamma_2)| +(15)^{\alpha}|\mathbf{E}_{15, 1}(\Gamma_2)| +(135)^{\alpha}|\mathbf{E}_{15, 9}(\Gamma_2)|+(225)^{\alpha}|\mathbf{E}_{15, 15}(\Gamma_2)|.$

After simplification, we get

$\mathfrak{R}\mathbb{R}_{\alpha}(\Gamma_2) = \left(\frac{3}{2}+6\, (15)^{\alpha}+3\, (225)^{\alpha} \right) {t}^{2}+ \left(\frac{-21}{2}+6\, (9)^{\alpha}+3\, (81)^{\alpha}-30\, (15)^{\alpha}+\right. \\ \left. 18\, (135)^{\alpha}-6\, (225)^{\alpha} \right) t+18-18\, (9)^{\alpha}-6\, (81)^ {\alpha}+36\, (15)^{\alpha}-30\, (135)^{\alpha}+9\, (225)^{\alpha}.$

Put $\alpha = 1$ and after some calculation, we get reverse second Zagreb index $\mathfrak{R}\mathbb{M}_2(\Gamma_2)$ as:

$\begin{equation} \mathfrak{R}\mathbb{M}_2(\Gamma_2) = {\frac {1533\, {t}^{2}}{2}}+{\frac {1833\, t}{2}}-2115 \end{equation}$

(4.2)

Using the Eq (1.4), we can determine the reverse first Zagreb index $\mathfrak{R}\mathbb{M}_1(\Gamma_2)$ as:

$\mathfrak{R}\mathbb{M}_1(\Gamma_2) = 2\times|\mathbf{E}_{1, 1}(\Gamma_1)|+10\times|\mathbf{E}_{9, 1}(\Gamma_1)| +18\times|\mathbf{E}_{9, 9}(\Gamma_1)|+16\times|\mathbf{E}_{15, 1}(\Gamma_1)| +24\times|\mathbf{E}_{15, 9}(\Gamma_1)|+30\times|\mathbf{E}_{15, 15}(\Gamma_1)|.$

By putting the values of from Eq (4.1) and after simplification, we obtain:

$\begin{equation} \mathfrak{R}\mathbb{M}_1(\Gamma_2) = 189\, {t}^{2}-135\, t-126 \end{equation}$

(4.3)

Using the Eq (1.6), we can determine the reverse hyper Zagreb index $\mathfrak{R}\mathbb{HM}(\Gamma_2)$ as:

$\mathfrak{R}\mathbb{HM}(\Gamma_2) = 4\times|\mathbf{E}_{1, 1}(\Gamma_2)|+100\times|\mathbf{E}_{9, 1}(\Gamma_2)| +324\times|\mathbf{E}_{9, 9}(\Gamma_2)|+256\times|\mathbf{E}_{15, 1}(\Gamma_2)| +576\times|\mathbf{E}_{15, 9}(\Gamma_2)|+900\times|\mathbf{E}_{15, 15}(\Gamma_2)|.$

After simplification, we get

$\begin{equation*} \mathfrak{R}\mathbb{HM}(\Gamma_2) = 4242\, {t}^{2}-1182\, t-3636. \end{equation*}$

Using the Eq (1.7), we can determine the reverse forgotten index $\mathfrak{R}\mathbb{F}(\Gamma_2))$ as:

$\mathfrak{R}\mathbb{F}(\Gamma_2)) = 2\times|\mathbf{E}_{1, 1}(\Gamma_2)|+82\times|\mathbf{E}_{9, 1}(\Gamma_2)| +162\times|\mathbf{E}_{9, 9}(\Gamma_2)|+226\times|\mathbf{E}_{15, 1}(\Gamma_2)| +306\times|\mathbf{E}_{15, 9}(\Gamma_2)|+450\times|\mathbf{E}_{15, 15}(\Gamma_2)|.$

After simplification, we get

$\begin{equation*} \mathfrak{R}\mathbb{F}(\Gamma_2) = 2709\, {t}^{2}-3015\, t+594. \end{equation*}$

In the next theorem, we will calculate the reverse atom bond connectivity index, reverse geometric arithmetic index for $\Gamma_2$ graph.

Theorem 4.2. Let $\Gamma_2$ be the third type of triangular hex-derived network, then

● $\mathfrak{R}\mathbb{ABC}(\Gamma_2) = \left(\frac{2\, \sqrt {210}}{5}+\frac{2\, \sqrt {7}}{5} \right) {t}^{2}+ \left(4\, \sqrt {2}+\frac{4}{3}-2\, \sqrt {210}+\frac{2\, \sqrt {330}}{5}-\frac{4\, \sqrt {7}}{5} \right) t -\frac{8}{3}-12\, \sqrt {2}+{\frac {12\, \sqrt {210}}{5}}-\frac{2\, \sqrt {330}}{3}+\frac{6\, \sqrt {7}}{5}$

● $\mathfrak{R}\mathbb{GA}(\Gamma_2) = \left(\frac{9}{2}+\frac{3\, \sqrt {15}}{4} \right) {t}^{2}+ \left(-{\frac{99}{10}}+\frac{3\, \sqrt {15}}{4} \right) t+{\frac{51}{5}}-3\, \sqrt {15}.$

Proof. The reverse atom bond connectivity ( $\mathfrak{R}\mathbb{ABC}(\Gamma_2)$ ), can be determined by using Eq (1.2) and Eq (4.1), as follows:

$\mathfrak{R}\mathbb{ABC}(\Gamma_2) = 0\times|\mathbf{E}_{1, 1}(\Gamma_2)|+\sqrt{\frac{8}{9}}\times|\mathbf{E}_{9, 1}(\Gamma_2)| +\sqrt{\frac{16}{81}}\times|\mathbf{E}_{9, 9}(\Gamma_2)|+\sqrt{\frac{14}{15}}\times|\mathbf{E}_{15, 1}(\Gamma_2)| +\sqrt{\frac{22}{135}}\times|\mathbf{E}_{15, 9}(\Gamma_2)|+\sqrt{\frac{28}{225}}\times|\mathbf{E}_{15, 15}(\Gamma_2)|.$

After some simplification, we get

$\mathfrak{R}\mathbb{ABC}(\Gamma_2) = \left(\frac{2\, \sqrt {210}}{5}+\frac{2\, \sqrt {7}}{5} \right) {t}^{2}+ \left(4\, \sqrt {2}+\frac{4}{3}-2\, \sqrt {210}+\frac{2\, \sqrt {330}}{5}-\frac{4\, \sqrt {7}}{5} \right) t -\frac{8}{3}-12\, \sqrt {2}+{\frac {12\, \sqrt {210}}{5}}-\frac{2\, \sqrt {330}}{3}+\frac{6\, \sqrt {7}}{5}.$

The reverse geometric arithmetic ( $\mathfrak{R}\mathbb{GA}(\Gamma_2)$ ), can be determined by using Eq (1.3) and Eq (4.1), as follows:

$\mathfrak{R}\mathbb{GA}(\Gamma_2) = |\mathbf{E}_{1, 1}(\Gamma_2)|+\frac{2\, \sqrt{9}}{10}\times|\mathbf{E}_{9, 1}(\Gamma_2)| +\frac{2\, \sqrt{81}}{18}\times|\mathbf{E}_{9, 9}(\Gamma_2)|+\frac{2\, \sqrt{15}}{16}\times|\mathbf{E}_{15, 1}(\Gamma_2)| +\frac{2\, \sqrt{135}}{24}\times|\mathbf{E}_{15, 9}(\Gamma_2)|+\frac{2\, \sqrt{225}}{30}\times|\mathbf{E}_{15, 15}(\Gamma_2)|.$

After some simplification, we get

$\mathfrak{R}\mathbb{GA}(\Gamma_2) = \left( \frac{9}{2}+\frac{3\, \sqrt {15}}{4} \right) {t}^{2}+ \left( -{\frac{99}{10}}+\frac{3\, \sqrt {15}}{4} \right) t+{\frac{51}{5}}-3\, \sqrt {15}.$

In the next theorems, we will calculate the reverse redefined Zagreb indices for $\Gamma_2$ graph.

Theorem 4.3. Let $\Gamma_2$ be the third type of triangular hex-derived network, then

● $\mathfrak{R}\mathbb{ReZ}_1(\Gamma_2) = {\frac {49\, {t}^{2}}{5}}-{\frac {649\, t}{15}}+{\frac{734}{15}}$

● $\mathfrak{R}\mathbb{ReZ}_2(\Gamma_2) = {\frac {231\, {t}^{2}}{8}}+{\frac {1671\, t}{40}}-{\frac{1017}{10}}$

● $\mathfrak{R}\mathbb{ReZ}_3(\Gamma_2) = 21693\, {t}^{2}+15513\, t-38142.$

Proof. Reverse redefined Zagreb indices can be calculated by using Eq (4.1), the $\mathfrak{R}\mathbb{ReZ}_1(\Gamma_2)$ by using Eq (1.9) as follows:

$\mathfrak{R}\mathbb{ReZ}_1(\Gamma_2) = 2\times|\mathbf{E}_{1, 1}(\Gamma_2)|+\frac{10}{9}\times|\mathbf{E}_{9, 1}(\Gamma_2)| +\frac{18}{81}\times|\mathbf{E}_{9, 9}(\Gamma_2)|+\frac{16}{15}\times|\mathbf{E}_{15, 1}(\Gamma_2)| +\frac{24}{135}\times|\mathbf{E}_{15, 9}(\Gamma_2)|+\frac{30}{225}\times|\mathbf{E}_{15, 15}(\Gamma_2)|.$

After some simplification, we get

$\mathfrak{R}\mathbb{ReZ}_1(\Gamma_2) = {\frac {49\, {t}^{2}}{5}}-{\frac {649\, t}{15}}+{\frac{734}{15}}.$

The $\mathfrak{R}\mathbb{ReZ}_2(\Gamma_2)$ can be determined by using Eq (1.10) as follows:

$\mathfrak{R}\mathbb{ReZ}_2(\Gamma_2) = \frac{1}{2}\times|\mathbf{E}_{1, 1}(\Gamma_2)|+\frac{9}{10}\times|\mathbf{E}_{9, 1}(\Gamma_2)| +\frac{81}{18}\times|\mathbf{E}_{9, 9}(\Gamma_2)|+\frac{15}{16}\times|\mathbf{E}_{15, 1}(\Gamma_2)| +\frac{135}{24}\times|\mathbf{E}_{15, 9}(\Gamma_2)|+\frac{225}{30}\times|\mathbf{E}_{15, 15}(\Gamma_2)|.$

After some simplification, we get

$\mathfrak{R}\mathbb{ReZ}_2(\Gamma_2) = {\frac {231\, {t}^{2}}{8}}+{\frac {1671\, t}{40}}-{\frac{1017}{10}}.$

The $\mathfrak{R}\mathbb{ReZ}_3(\Gamma_2)$ can be calculated by using Eq (1.11) as follows:

$\mathfrak{R}\mathbb{ReZ}_3(\Gamma_2) = 2\times|\mathbf{E}_{1, 1}(\Gamma_2)|+90\times|\mathbf{E}_{9, 1}(\Gamma_2)| +1458\times|\mathbf{E}_{9, 9}(\Gamma_2)|+240\times|\mathbf{E}_{15, 1}(\Gamma_2)| +3240\times|\mathbf{E}_{15, 9}(\Gamma_2)|+6750\times|\mathbf{E}_{15, 15}(\Gamma_2)|.$

After some simplification, we get

$\mathfrak{R}\mathbb{ReZ}_3(\Gamma_2) = 21693\, {t}^{2}+15513\, t-38142.$

5. Rectangular hex-derived network $RHDN3(p)$

In this section, we calculate certain reverse degree based topological indices of the third type of rectangular hex-derived network, $RHDN_3(t, w)$ of dimension $t = w.$ Now we calculated reverse degree based indices such as: reverse Randic index $\mathfrak{R}\mathbb{R}_{\alpha},$ reverse atom bond connectivity index $\mathfrak{R}\mathbb{ABC},$ reverse geometric arithmetic index $\mathfrak{R}\mathbb{GA},$ first reverse Zagreb index $\mathfrak{R}\mathbb{M}_1,$ second reverse Zagreb index $\mathfrak{R}\mathbb{M}_2$ , reverse forgotten index $\mathfrak{R}\mathbb{F},$ reverse augmented Zagreb index $\mathfrak{R}\mathbb{AZI}$ , reverse hyper Zagreb index $\mathfrak{R}\mathbb{HM}$ and reverse redefined Zagreb indices for $\Gamma_3$ graph.

$\begin{equation} |\mathbf{E}_{\mathfrak{R}(\omega), \mathfrak{R}(\mu)}(\Gamma_3)| = \left\{ \begin{array}{ll} 3t^2-16t+21, & \hbox{for;}\ \mathfrak{R}(\omega) = 1, \mathfrak{R}(\mu) = 1 \\ 8t-20, & \hbox{for;}\ \mathfrak{R}(\omega) = 9, \mathfrak{R}(\mu) = 1 \\ 4t-10, & \hbox{for;}\ \mathfrak{R}(\omega) = 9, \mathfrak{R}(\mu) = 9 \\ 2, & \hbox{for;}\ \mathfrak{R}(\omega) = 12, \mathfrak{R}(\mu) = 1 \\ 4, & \hbox{for;}\ \mathfrak{R}(\omega) = 12, \mathfrak{R}(\mu) = 9 \\ 12t^2-48t+48, & \hbox{for;}\ \mathfrak{R}(\omega) = 15, \mathfrak{R}(\mu) = 1 \\ 24t-44, & \hbox{for;}\ \mathfrak{R}(\omega) = 15, \mathfrak{R}(\mu) = 9 \\ 8, & \hbox{for;}\ \mathfrak{R}(\omega) = 15, \mathfrak{R}(\mu) = 12 \\ 6t^2-12t+10, & \hbox{for;}\ \mathfrak{R}(\omega) = 15, \mathfrak{R}(\mu) = 15 \end{array} \right. \end{equation}$

(5.1)

In the next theorem, we will calculate the reverse Randic index, reverse Zagreb indices, reverse forgotten index for $\Gamma_3$ graph.

Theorem 5.1. Let $\Gamma_3$ be the third type of rectangular hex-derived network, then

● $\mathfrak{R}\mathbb{R}_{\alpha}(\Gamma_3) = \left(3+12\, (15)^{\alpha}+6\, (225)^{\alpha} \right) {t}^{2}+ \left( -16+8\, (9)^{\alpha}+4\, (81)^{\alpha}-48\, (15)^{\alpha}+24\, (135)^{ \alpha}-12\, (225)^{\alpha} \right) t+21-20\, (9)^{\alpha}-10\, (81)^{ \alpha}+2\, (12)^{\alpha}+4\, (108)^{\alpha}+48\, (15)^{\alpha}-44\, (135) ^{\alpha}+8\, (180)^{\alpha}+10\, (225)^{\alpha}$

● $\mathfrak{R}\mathbb{M}_1(\Gamma_3) = 378\, {t}^{2}-432\, t$

● $\mathfrak{R}\mathbb{M}_2(\Gamma_3) = 1533\, {t}^{2}+200\, t-2043$

● $\mathfrak{R}\mathbb{HM}(\Gamma_3) = 8484\, {t}^{2}-7232\, t-1278$

● $\mathfrak{R}\mathbb{F}(\Gamma_3) = 5418\, {t}^{2}-7632\, t+2808.$

Proof. Let $\Gamma_3$ be the third type of rectangular hex-derived network which is shown in . The order of hexagonal hex derived network $\Gamma_3$ is $p = |\Gamma_1| = 7t^2 -12t + 6$ and size is $q = 21t^2-40t+19.$ The edge partitioned of $\Gamma_3$ based on their reverse degrees are shown in Eq (5.1). Reverse Randic index can be calculated by using Eq (5.1). Thus, from Eq (1.1), it follows,

$\mathfrak{R}\mathbb{R}_{\alpha}(\Gamma_3) = \left.(1)^{\alpha}|\mathbf{E}_{1, 1}(\Gamma_3)|+(9)^{\alpha}|\mathbf{E}_{9, 1}(\Gamma_3)| +(81)^{\alpha}|\mathbf{E}_{9, 9}(\Gamma_3)|+(12)^{\alpha}|\mathbf{E}_{12, 1}(\Gamma_3)|\right.\\ +(108)^{\alpha}|\mathbf{E}_{12, 9}(\Gamma_3)| +(15)^{\alpha}|\mathbf{E}_{15, 1}(\Gamma_3)| +(135)^{\alpha}|\mathbf{E}_{15, 9}(\Gamma_3)|+(180)^{\alpha}|\mathbf{E}_{15, 12}(\Gamma_3)| +(225)^{\alpha}|\mathbf{E}_{15, 15}(\Gamma_3)|.$

After Simplification, we get

$\mathfrak{R}\mathbb{R}_{\alpha}(\Gamma_3) = \left(3+12\, (15)^{\alpha}+6\, (225)^{\alpha} \right) {t}^{2}+ \left(-16+8\, (9)^{\alpha}+4\, (81)^{\alpha}-48\, (15)^{\alpha}+24\, (135)^{ \alpha}-12\, (225)^{\alpha} \right) t+21-20\, (9)^{\alpha}-10\, (81)^{ \alpha}+2\, (12)^{\alpha}+4\, (108)^{\alpha}+48\, (15)^{\alpha}-44\, (135) ^{\alpha}+8\, (180)^{\alpha}+10\, (225)^{\alpha}.$

Put $\alpha = 1$ and after some calculation, we get reverse second Zagreb index $\mathfrak{R}\mathbb{M}_2(\Gamma_3)$ as:

$\begin{equation} \mathfrak{R}\mathbb{M}_2(\Gamma_3) = 1533\, {t}^{2}+200\, t-2043. \end{equation}$

(5.2)

Using the Eq (1.4), we can determine the reverse first Zagreb index $\mathfrak{R}\mathbb{M}_1(\Gamma_3)$ as:

$\mathfrak{R}\mathbb{M}_1(\Gamma_3) = 2\times|\mathbf{E}_{1, 1}(\Gamma_3)|+10\times|\mathbf{E}_{9, 1}(\Gamma_3)| +18\times|\mathbf{E}_{9, 9}(\Gamma_3)|+13\times|\mathbf{E}_{12, 1}(\Gamma_3)|+21\times|\mathbf{E}_{12, 9}(\Gamma_3)| +16\times|\mathbf{E}_{15, 1}(\Gamma_3)| +24\times|\mathbf{E}_{15, 9}(\Gamma_3)|+27\times|\mathbf{E}_{15, 12}(\Gamma_3)| +30\times|\mathbf{E}_{15, 15}(\Gamma_3)|.$

By putting the values of from Eq (5.1) and after simplification, we obtain:

$\begin{equation} \mathfrak{R}\mathbb{M}_1(\Gamma_3) = 378\, {t}^{2}-432\, t. \end{equation}$

(5.3)

Using the Eq (1.6), we can determine the reverse hyper Zagreb index $\mathfrak{R}\mathbb{HM}(\Gamma_3)$ as:

$\mathfrak{R}\mathbb{HM}(\Gamma_3) = 4\times|\mathbf{E}_{1, 1}(\Gamma_3)|+100\times|\mathbf{E}_{9, 1}(\Gamma_3)| +324\times|\mathbf{E}_{9, 9}(\Gamma_3)|+169\times|\mathbf{E}_{12, 1}(\Gamma_3)|+441\times|\mathbf{E}_{12, 9}(\Gamma_3)| +256\times|\mathbf{E}_{15, 1}(\Gamma_3)| +576\times|\mathbf{E}_{15, 9}(\Gamma_3)|+729\times|\mathbf{E}_{15, 12}(\Gamma_3)| +900\times|\mathbf{E}_{15, 15}(\Gamma_3)|.$

After simplification, we get

$\begin{equation*} \mathfrak{R}\mathbb{HM}(\Gamma_3) = 8484\, {t}^{2}-7232\, t-1278. \end{equation*}$

Using the Eq (1.7), we can determine the reverse forgotten index $\mathfrak{R}\mathbb{F}(\Gamma_3))$ as:

$\mathfrak{R}\mathbb{F}(\Gamma_3)) = 2\times|\mathbf{E}_{1, 1}(\Gamma_3)|+82\times|\mathbf{E}_{9, 1}(\Gamma_3)| +162\times|\mathbf{E}_{9, 9}(\Gamma_3)|+145\times|\mathbf{E}_{12, 1}(\Gamma_3)|+225\times|\mathbf{E}_{12, 9}(\Gamma_3)| +226\times|\mathbf{E}_{15, 1}(\Gamma_3)| +306\times|\mathbf{E}_{15, 9}(\Gamma_3)|+369\times|\mathbf{E}_{15, 12}(\Gamma_3)| +450\times|\mathbf{E}_{15, 15}(\Gamma_3)|.$

After simplification, we get

$\begin{equation*} \mathfrak{R}\mathbb{F}(\Gamma_3) = 5418\, {t}^{2}-7632\, t+2808. \end{equation*}$

In the next theorem, we will calculate the reverse atom bond connectivity index, reverse geometric arithmetic index for $\Gamma_3$ graph.

Theorem 5.2. Let $\Gamma_3$ be the third type of rectangular hex-derived network, then

● $\mathfrak{R}\mathbb{ABC}(\Gamma_3) = \left(\frac{4\, \sqrt {210}}{5}+\frac{4\, \sqrt {7}}{5} \right) {t}^{2}+ \left(\frac{16\, \sqrt {2}}{3}+{\frac{16}{9}}-{\frac {16\, \sqrt {210}}{5}}+{\frac {8\, \sqrt {330}}{15}}-\frac{8\, \sqrt {7}}{5} \right) t-{\frac{40}{9}}-{\frac {40\, \sqrt {2}}{3}}+\frac{\sqrt {33}}{3}+\frac{2\, \sqrt {57}}{9}+{\frac {16\, \sqrt {210} }{5}}-{\frac {44\, \sqrt {330}}{45}}+\frac{4\, \sqrt {5}}{3}+\frac{4\, \sqrt {7}}{3}.$

● $\mathfrak{R}\mathbb{GA}(\Gamma_3) = \left(9+\frac{3\, \sqrt {15}}{2} \right) {t}^{2}-{\frac {96\, t}{5}}+9+{\frac {264\, \sqrt {3}}{91}}-5\, \sqrt {15}+{\frac {32\, \sqrt {5}}{9}}.$

Proof. The reverse atom bond connectivity ( $\mathfrak{R}\mathbb{ABC}(\Gamma_3)$ ), can be determined by using Eq (1.2) and Eq (5.1), as follows:

$\mathfrak{R}\mathbb{ABC}(\Gamma_3) = 0\times|\mathbf{E}_{1, 1}(\Gamma_3)|+\sqrt{\frac{8}{9}}\times|\mathbf{E}_{9, 1}(\Gamma_3)| +\sqrt{\frac{16}{81}}\times|\mathbf{E}_{9, 9}(\Gamma_3)|+\sqrt{\frac{11}{12}}\times|\mathbf{E}_{12, 1}(\Gamma_3)| +\sqrt{\frac{19}{108}}\times|\mathbf{E}_{12, 9}(\Gamma_3)| +\sqrt{\frac{14}{15}}\times|\mathbf{E}_{15, 1}(\Gamma_3)| +\sqrt{\frac{22}{135}}\times|\mathbf{E}_{15, 9}(\Gamma_3)|+\sqrt{\frac{25}{180}}\times|\mathbf{E}_{15, 12}(\Gamma_3)| +\sqrt{\frac{28}{225}}\times|\mathbf{E}_{15, 15}(\Gamma_3)|.$

After some simplification, we get

$\mathfrak{R}\mathbb{ABC}(\Gamma_3) = \left(\frac{4\, \sqrt {210}}{5}+\frac{4\, \sqrt {7}}{5} \right) {t}^{2}+ \left(\frac{16\, \sqrt {2}}{3}+{\frac{16}{9}}-{\frac {16\, \sqrt {210}}{5}}+{\frac {8\, \sqrt {330}}{15}}-\frac{8\, \sqrt {7}}{5} \right) t-{\frac{40}{9}}-{\frac {40\, \sqrt {2}}{3}}+\frac{\sqrt {33}}{3}+\frac{2\, \sqrt {57}}{9}+{\frac {16\, \sqrt {210} }{5}}-{\frac {44\, \sqrt {330}}{45}}+\frac{4\, \sqrt {5}}{3}+\frac{4\, \sqrt {7}}{3}.$

The reverse geometric arithmetic ( $\mathfrak{R}\mathbb{GA}(\Gamma_3)$ ), can be determined by using Eq (1.3) and Eq (5.1), as follows:

$\mathfrak{R}\mathbb{GA}(\Gamma_3) = |\mathbf{E}_{1, 1}(\Gamma_3)|+\frac{2\, \sqrt{9}}{10}\times|\mathbf{E}_{9, 1}(\Gamma_3)| +\frac{2\, \sqrt{81}}{18}\times|\mathbf{E}_{9, 9}(\Gamma_3)|+\frac{2\, \sqrt{12}}{13}\times|\mathbf{E}_{12, 1}(\Gamma_3)| +\frac{2\, \sqrt{108}}{21}\times|\mathbf{E}_{12, 9}(\Gamma_3)| +\frac{2\, \sqrt{15}}{16}\times|\mathbf{E}_{15, 1}(\Gamma_3)| +\frac{2\, \sqrt{135}}{24}\times|\mathbf{E}_{15, 9}(\Gamma_3)|+\frac{2\, \sqrt{180}}{27}\times|\mathbf{E}_{15, 12}(\Gamma_3)| +\frac{2\, \sqrt{225}}{30}\times|\mathbf{E}_{15, 15}(\Gamma_3)|.$

After some simplification, we get

$\mathfrak{R}\mathbb{GA}(\Gamma_3) = \left(9+\frac{3\, \sqrt {15}}{2} \right) {t}^{2}-{\frac {96\, t}{5}}+9+{\frac {264\, \sqrt {3}}{91}}-5\, \sqrt {15}+{\frac {32\, \sqrt {5}}{9}}.$

In the next theorems, we will calculate the reverse redefined Zagreb indices for $\Gamma_3$ graph.

Theorem 5.3. Let $\Gamma_3$ be the third type of rectangular hex-derived network, then

● $\mathfrak{R}\mathbb{ReZ}_1(\Gamma_3) = {\frac {98\, {t}^{2}}{5}}-{\frac {3184\, t}{45}}+{\frac{5977}{90}}$

● $\mathfrak{R}\mathbb{ReZ}_2(\Gamma_3) = {\frac {231\, {t}^{2}}{4}}+{\frac {86\, t}{5}}-{\frac{28460}{273}}$

● $\mathfrak{R}\mathbb{ReZ}_3(\Gamma_3) = 43386\, {t}^{2}-8240\, t-31614.$

Proof. Reverse redefined Zagreb indices can be calculated by using Eq (5.1), the $\mathfrak{R}\mathbb{ReZ}_1(\Gamma_3)$ by using Eq (1.9) as follows:

$\mathfrak{R}\mathbb{ReZ}_1(\Gamma_3) = 2\times|\mathbf{E}_{1, 1}(\Gamma_3)|+\frac{10}{9}\times|\mathbf{E}_{9, 1}(\Gamma_3)| +\frac{18}{81}\times|\mathbf{E}_{9, 9}(\Gamma_3)|+\frac{13}{12}\times|\mathbf{E}_{12, 1}(\Gamma_3)| +\frac{21}{108}\times|\mathbf{E}_{12, 9}(\Gamma_3)| +\frac{16}{15}\times|\mathbf{E}_{15, 1}(\Gamma_3)| +\frac{24}{135}\times|\mathbf{E}_{15, 9}(\Gamma_3)|+\frac{27}{108}\times|\mathbf{E}_{15, 12}(\Gamma_3)| +\frac{30}{225}\times|\mathbf{E}_{15, 15}(\Gamma_3)|.$

After some simplification, we get

$\mathfrak{R}\mathbb{ReZ}_1(\Gamma_3) = {\frac {98\, {t}^{2}}{5}}-{\frac {3184\, t}{45}}+{\frac{5977}{90}}.$

The $\mathfrak{R}\mathbb{ReZ}_2(\Gamma_1)$ can be determined by using equation (1.10) as follows:

$\mathfrak{R}\mathbb{ReZ}_2(\Gamma_3) = \frac{1}{2}\times|\mathbf{E}_{1, 1}(\Gamma_3)|+\frac{9}{10}\times|\mathbf{E}_{9, 1}(\Gamma_3)| +\frac{81}{18}\times|\mathbf{E}_{9, 9}(\Gamma_3)|+\frac{12}{13}\times|\mathbf{E}_{12, 1}(\Gamma_3)| +\frac{108}{21}\times|\mathbf{E}_{12, 9}(\Gamma_3)| +\frac{15}{16}\times|\mathbf{E}_{15, 1}(\Gamma_3)| +\frac{135}{24}\times|\mathbf{E}_{15, 9}(\Gamma_3)|+\frac{108}{27}\times|\mathbf{E}_{15, 12}(\Gamma_3)| +\frac{225}{30}\times|\mathbf{E}_{15, 15}(\Gamma_3)|.$

After some simplification, we get

$\mathfrak{R}\mathbb{ReZ}_2(\Gamma_3) = {\frac {231\, {t}^{2}}{4}}+{\frac {86\, t}{5}}-{\frac{28460}{273}}.$

The $\mathfrak{R}\mathbb{ReZ}_3(\Gamma_3)$ can be calculated by using Eq (1.11) as follows:

$\mathfrak{R}\mathbb{ReZ}_3(\Gamma_3) = 2\times|\mathbf{E}_{1, 1}(\Gamma_3)|+90\times|\mathbf{E}_{9, 1}(\Gamma_3)| +1458\times|\mathbf{E}_{9, 9}(\Gamma_3)|+156\times|\mathbf{E}_{12, 1}(\Gamma_3)| +2268\times|\mathbf{E}_{12, 9}(\Gamma_3)| +240\times|\mathbf{E}_{15, 1}(\Gamma_3)| +3240\times|\mathbf{E}_{15, 9}(\Gamma_3)|+2916\times|\mathbf{E}_{15, 12}(\Gamma_3)| +6750\times|\mathbf{E}_{15, 15}(\Gamma_3)|.$

After some simplification, we get

$\mathfrak{R}\mathbb{ReZ}_3(\Gamma_3) = 43386\, {t}^{2}-8240\, t-31614.$

6. Conclusions

In this article, we have calculated the exact solutions of reverse degree-based topological descriptors for hex-derived networks of third type. Hex-derived network has a variety of useful applications in pharmacy, electronics, and networking. We obtained the reverse degree-based indices such as reverse Randic index, reverse atom bond connectivity index, reverse geometric arithmetic index, reverse Zagreb indices, reverse redefined Zagreb indices for hex derived networks. These results may be helpful for people working in computer science and chemistry who encounter hex-derived networks.

Conflict of interest

The authors declare that there is no conflict of financial interests regarding the publication of this paper.

References

[1]	Z. Wang, X. Liu, Lightweight optimization design of arm frame for aerial work platform based on NSGA-Ⅱ algorithm, IOP Conf. Ser.: Mater. Sci. Eng., 733 (2020), 012037. https://doi.org/10.1088/1757-899X/733/1/012037 doi: 10.1088/1757-899X/733/1/012037
[2]	G. Sun, H. Zhang, J. Fang, G. Li, Q. Li, Multi-objective and multi-case reliability-based design optimization for tailor rolled blank (TRB) structures, Struct. Multidiscip. Optim., 55 (2017), 1899-1916. https://doi.org/10.1007/s00158-016-1592-1 doi: 10.1007/s00158-016-1592-1
[3]	L. Duan, H. Jiang, A. Cheng, H. Xue, G. Geng, Multi-objective reliability-based design optimization for the VRB-VCS FLB under front-impact collision, Struct. Multidiscip. Optim., 59 (2019), 1835-1851. https://doi.org/10.1007/s00158-018-2142-9 doi: 10.1007/s00158-018-2142-9
[4]	N. Qiu, Y. Gao, J. Fang, G. Sun, N. H. Kim, Topological design of multi-cell hexagonal tubes under axial and lateral loading cases using a modified particle swarm algorithm, Appl. Math. Modell., 53 (2018), 567-583. https://doi.org/10.1016/j.apm.2017.08.017 doi: 10.1016/j.apm.2017.08.017
[5]	A. Y. Ismail, G. Na, B. Koo, Topology and response surface optimization of a bicycle crank arm with multiple load cases, Appl. Sci., 10 (2020), 2201. https://doi.org/10.3390/app10062201 doi: 10.3390/app10062201
[6]	Y. X. Du, J. R. Hu, Z. F. Fang, Q. H. Tian, Topology optimization for work flat of tower-belt under multi-loading cases, Appl. Mech. Mater., 197 (2012), 614-618. https://doi.org/10.4028/www.scientific.net/AMM.197.614 doi: 10.4028/www.scientific.net/AMM.197.614
[7]	Y. Wang, H. T. Zhu, L. Xu, J. P. Ji, X. L. Dai, Study on optimization of engine accessory bracket based on optistruct, Adv. Mater. Res., 421 (2012), 308-311. https://doi.org/10.4028/www.scientific.net/AMR.421.308 doi: 10.4028/www.scientific.net/AMR.421.308
[8]	C. Zhuang, Z. Xiong, H. Ding, A level set method for topology optimization of heat conduction problem under multiple load cases, Comput. Methods Appl. Mech. Eng., 196 (2007), 1074-1084. https://doi.org/10.1016/j.cma.2006.08.005 doi: 10.1016/j.cma.2006.08.005
[9]	L. Duan, N. Xiao, Z. Hu, G. Li, A. Cheng, An efficient lightweight design strategy for body-in-white based on implicit parameterization technique, Struct. Multidiscip. Optim., 55 (2017), 1927-1943. https://doi.org/10.1007/s00158-016-1621-0 doi: 10.1007/s00158-016-1621-0
[10]	B. Mi, S. Cheng, Y. Luo, H. Fan, A new many-objective aerodynamic optimization method for symmetrical elliptic airfoils by PSO and direct-manipulation-based parametric mesh deformation, Aerosp. Sci. Technol., 120 (2022), 107296. https://doi.org/10.1016/j.ast.2021.107296 doi: 10.1016/j.ast.2021.107296
[11]	W. Zhang, S. Feng, Combined parameterization of material distribution and surface mesh for stiffener layout optimization of complex surfaces, Struct. Multidiscip. Optim., 65 (2022), 103. https://doi.org/10.1007/s00158-022-03191-3 doi: 10.1007/s00158-022-03191-3
[12]	C. Zhang, S. Yang, Q. Zhang, G. Zhao, P. Lu, W. Sun, Automatic optimization design of a feeder extrusion die with response surface methodology and mesh deformation technique, Int. J. Adv. Manuf. Technol., 91 (2017), 3181-3193. https://doi.org/10.1007/s00170-017-0018-6 doi: 10.1007/s00170-017-0018-6
[13]	D. Wang, R. Jiang, Y. Wu, A hybrid method of modified NSGA-Ⅱ and TOPSIS for lightweight design of parameterized passenger car sub-frame, J. Mech. Sci. Technol., 30 (2016), 4909-4917. https://doi.org/10.1007/s12206-016-1010-z doi: 10.1007/s12206-016-1010-z
[14]	A. Nazemian, P. Ghadimi, Shape optimisation of trimaran ship hull using CFD-based simulation and adjoint solver, Ships Offshore Struct., 17 (2022), 359-373. https://doi.org/10.1080/17445302.2020.1827807 doi: 10.1080/17445302.2020.1827807
[15]	W. Hou, C. Shan, P. Hu, H. Zhang, Multilevel optimisation method for vehicle body in conceptual design, Int. J. Veh. Des., 73 (2017), 157-178. https://doi.org/10.1504/IJVD.2017.082589 doi: 10.1504/IJVD.2017.082589
[16]	I. Gandikota, M. Rais-Rohani, M. Kiani, S. DorMohammadi, Multilevel design optimisation of a vehicle-dummy model under crash, vibration and injury criteria, Int. J. Veh. Des., 70 (2015), 45-68. https://doi.org/10.1504/IJVD.2016.073703
[17]	X. Liu, W. He, F. Wei, Design of high altitude propeller using multilevel optimization, Int. J. Comput. Methods, 17 (2020), 1950004. https://doi.org/10.1142/S021987621950004X doi: 10.1142/S021987621950004X
[18]	M. P. Bendsøe, N. Kikuchi, Generating optimal topologies in structural design using a homogenization method, Comput. Methods Appl. Mech. Eng., 71 (1988), 197-224. https://doi.org/10.1016/0045-7825(88)90086-2 doi: 10.1016/0045-7825(88)90086-2

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