On valency based topological properties of the starphene $\mathit{St}[n,m,l]$ and fenestrene $F[n,m]$

1 Introduction

Chemical graph theory is a branch of mathematical chemistry that applies graph theory to mathematical modeling of chemical phenomena. The topological index is a graph invariant that describes the molecular structure’s topology and transforms it to a real number that predicts certain of its physico-chemical characteristics such as freezing point, infrared spectrum, boiling point, viscosity, melting point, electronic parameters, and density (Zhong et al., 2021; Süleyman, 2017; Rauf et al., 2021). The reader may access to some of the previously published works on topological indices (Rauf et al., 2022; Sarkar et al., 2018; Sarkar and Pal, 2020; Naeem and Rauf, 2021; Rauf et al., 2021).

Starphene is a single ring of benzene that was surrounded by three identical substituents of arene (Rüdiger et al., 2019). The derivatives of benzene are manufactured on large scale for use in high-quality octane gasoline and production of polymer. Ortho-bromotetracene and ortho-dibromopentacene are suitable and undergoes the trimerization of Ni-catalyzed for the formation of starphene. The red starphene is made up of three hexacenes, with one centre ring shared with the orange star. Decastarphene-(3,3,3) is the largest unsubstituted derivatives of starphene which is formulated in 1968 by Clar and Mullen (1968). It is very complicated to prove that this is a complete aromatic conjugation that does not exist in starphene substituent. Starphene is an attractive compound due to its physicochemical properties. They inherited their character from arenes, which make them an interesting compound for the study of organic electronics and optics. It makes it very challenging for the synthesis of larger starphene due to their instability (Holec et al., 2021). The general fenestrenes uses the following formulas: $$L\left(\left[\begin{array}{ccccc}\hfill \underset{\frac{m-1}{2}-2}{\underbrace{2,\dots ,2}}\hfill & \hfill ,n,\hfill & \hfill \underset{m-1}{\underbrace{2,2,\dots ,2}}\hfill & \hfill ,n,\hfill & \hfill \underset{\frac{m-1}{2}-2}{\underbrace{2,\dots ,2}}\hfill \end{array}\right]\right)\text{.}$$ This is the ZZ polynomial of fenestrenes. This formula is helpful to find the alternative formula for the ZZ polynomial of fenestrenes $F(n,m)$ . The structure of fenestrenes contains zigzag chains that forms four disconnected parts (Chou and Witek, 2015). Generally, fenestrenes are used for the testing of pro-apoptotic activities on the two human cancer cell line (THP-1 and SW620) (Hulot et al., 2010).

Let $G=(V,E)$ be a simple graph, where $V=V\left(G\right)$ is a non-empty set of elements referred to as vertices or points and $E=E\left(G\right)$ is a set of unordered pairings of different members of $V\left(G\right)$ referred to as edges or lines. The sets $V\left(G\right)$ and $E\left(G\right)$ are referred to as the vertex set and edge set respectively. The degree of a vertex $v\in V\left(G\right)$ , represented by the symbol $d_v$ , is the number of edges that are incident to v. Two vertices u and v of G are adjacent if they are connected by an edge $e=\mathit{uv}$ . The open neighbourhood of v is defined as the collection of vertices that are connected to u and is represented by the symbol $N\left(u\right)$ . The closed neighbourhood of u denoted by $N\left[u\right]$ is obtained by adding the vertex u to $N\left(u\right)$ .

Topological indices (TI) plays an important role in explaining and quantifying the molecular structure of hydrocarbons. The first topological index was defined by Wiener (1947) while he was working on the boiling point of paraffin. Randic index was introduced by Randic (1975) and for any graph G, it is defined as $$R_{\frac{-1}{2}}\left(G\right)={\displaystyle \sum _{\mathit{uv}\in E}}\frac{1}{\sqrt{d_v{d}_u}}\text{.}$$ Gutman and Trinajstic (1972) proposed the first and second Zagreb index in 1972 and applied them to the branching problem (Gutman et al., 1975). The first and second Zagreb indices are defined as $$M_1\left(G\right)={\displaystyle \sum _{\mathit{uv}\in E\left(G\right)}}(d_v+d_u)$$ $$M_2\left(G\right)={\displaystyle \sum _{\mathit{uv}\in E\left(G\right)}}(d_v\times d_u)\text{.}$$ The sum connectivity index was proposed by Zhou and Trinajstic (2009). It is denoted and defined as: $$\mathit{SCI}\left(G\right)={\displaystyle \sum _{\mathit{xy}\in E}}\frac{1}{\sqrt{d_v+d_u}}\text{.}$$ Zhong (2012) proposed the harmonic index which is defined as follows: $$H\left(G\right)={\displaystyle \sum _{\mathit{uv}\in E}}\left(\frac{2}{d_v+d_u}\right)\text{.}$$ Estrada et al. (1998) introduced the notion of Atom Bond Connectivity (ABC) index which is defined as $$\mathit{ABC}\left(G\right)={\displaystyle \sum _{\mathit{uv}\in E}}\sqrt{\frac{d_v+d_u-2}{d_v{d}_u}}\text{.}$$ Graovac and Pisanski (1991) proposed Geometric Arithmetic (GA) index which is defined as $$\mathit{GA}\left(G\right)={\displaystyle \sum _{\mathit{uv}\in E}}\frac{2\sqrt{d_v{d}_u}}{d_v+d_u}\text{.}$$ For more details on the study and computation of topological indices, the reader may refer to (Aslam et al., 2018; Aslam et al., 2017; Bashir et al., 2017; Chu et al., 2021).

Recently, two variants of degree based topological indices were proposed by Randic (1975). These indices are based on ve-degree and eV degree. Some mathematical properties of these indices were discussed by Bollobs and Erdos (1998). Ediz (Amic et al., 1998; Graovac and Pisanski, 1991; Estrada et al., 1998) translated the classical degree based topological indices into eV-degree and ve-degree based topological indices. It was observed that ve-degree based zagreb index has strong predictability power than the classical Zagreb index. The eV-degree of any edge $\mathit{uv}=e\in E\left(G\right)$ is the total number of the vertices of closed neighbourhoods of the end vertices of an edge e. The eV degree of an edge e is denoted by ${\mathrm{\Psi }}_{\mathit{ev}}\left(e\right)$ . The ve-degree of any vertex $v\in V\left(G\right)$ is the total number of different edges which are adjacent to v and the first neighbor of v, and is denoted by ${\mathrm{\Psi }}_{\mathit{ve}}\left(v\right)$ . The Mathematical formulas for some of the eV-degree and ve-degree based indices are presented in Table 1.

2 Structure of starphene $\mathit{St}[n,m,l]$

In this section, we discuss a class of benzenoid system named starphene. Starphene is colorless and its melting point is 198°C. If three hexagons are attached by a common vertex, the benzenoid structure is called peri-condensed, otherwise, we say catacondensed. We denote the structure of starphene by $\mathit{St}[n,m,l]$ , where $m,n,l$ denotes the number of hexagons in each linear chain attached with the central hexagon respectively. The molecular structure of starphene is depicted in Fig. 1. The structure of $\mathit{St}[n,m,l]$ has total number of $2(2m+2n+2l-3)$ vertices and $5m+5n+5l-9$ edges. Let $V_i$ denotes the vertex set of $\mathit{St}[n,m,l]$ containing the vertices of degree i. Then the vertex set $V\left(\mathit{St}\right[n,m,l\left]\right)$ can be partitioned into two sets $V_2$ and $V_3$ with $\left|V_2\right|=2m+2n+2l$ and $\left|V_3\right|=2m+2n+2l-6$ . Let ${\epsilon }_{(i,j)}$ denotes the edge set containing the edges of $\mathit{St}[n,m,l]$ having degree of end vertices as i and j. The edge set of $\mathit{St}[n,m,l]$ can be partitioned as follows: ${\epsilon }_{(2,2)}$ containing 9 edge, ${\epsilon }_{(2,3)}$ containing $2(2m+2n+2l-9)$ edges and ${\epsilon }_{(3,3)}$ containing $m+n+l$ edges.

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Fig. 1

Structure of starphene $\mathit{St}[n,m,l]$ .

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2.1

2.1 Main results

Theorem 1

Let H denoted the graph of starphene $\mathit{St}[n,m,l]$ . Let $m,n,l⩾3$ , then eV-degree Zagreb and Randic index are

(a)   ${\mathsf{M}}^{\mathit{ev}}\left(H\right)=136m+136n+136l-306$ .

(b)   ${\mathsf{R}}^{\mathit{ev}}\left(H\right)=(\frac{4}{\sqrt{5}}+\frac{1}{\sqrt{6}})m+(\frac{4}{\sqrt{5}}+\frac{1}{\sqrt{6}})n+(\frac{4}{\sqrt{5}}+\frac{1}{\sqrt{6}})l+(\frac{9}{2}-\frac{18}{\sqrt{5}})$ .

Proof 1

By using the edge partition given in Table 2, we can compute the eV-degree Zagreb and Randić index as follows

Table 2 Edges partition for eV-degree of $\mathit{St}[n,m,l]$ .

$\left(\mathrm{\Psi }\right(u),\mathrm{\Psi }(v\left)\right)$	${\mathrm{\Psi }}_{\mathit{ev}}\left(e\right)$	Frequency
${\epsilon }_{(2,2)}$	4	9
${\epsilon }_{(2,3)}$	5	$2(2m+2n+2l-9)$
${\epsilon }_{(3,3)}$	6	$m+n+l$

(a) Ev-degree Zagreb index $$\begin{array}{cc}\hfill {\mathsf{M}}^{\mathit{ev}}\left(H\right)=& {\displaystyle \sum _{e\in E\left(H\right)}}{\mathrm{\Psi }}_{\mathit{ev}}{\left(e\right)}^2,\hfill \\ \hfill {\mathsf{M}}^{\mathit{ev}}\left(H\right)=& {\left(4\right)}^2\left|{\epsilon }_{(2,2)}\right|+{\left(5\right)}^2\left|{\epsilon }_{(2,3)}\right|+{\left(6\right)}^2\left|{\epsilon }_{(3,3)}\right|\hfill \\ \hfill =& 136m+136n+136l-306\text{.}\hfill \end{array}$$

(b) Ev-degree Randić index $$\begin{array}{cc}\hfill {\mathsf{R}}^{\mathit{ev}}\left(H\right)=& {\displaystyle \sum _{e\in E\left(H\right)}}{\mathrm{\Psi }}_{\mathit{ev}}{\left(e\right)}^{-\frac{1}{2}},\hfill \\ \hfill {\mathsf{R}}^{\mathit{ev}}\left(H\right)=& {\left(4\right)}^{-\frac{1}{2}}\left|{\epsilon }_{(4,2)}\right|+{\left(5\right)}^{-\frac{1}{2}}\left|{\epsilon }_{(2,3)}\right|+{\left(6\right)}^{-\frac{1}{2}}\left|{\epsilon }_{(3,3)}\right|\hfill \\ \hfill =& (\frac{4}{\sqrt{5}}+\frac{1}{\sqrt{6}})m+(\frac{4}{\sqrt{5}}+\frac{1}{\sqrt{6}})n+(\frac{4}{\sqrt{5}}+\frac{1}{\sqrt{6}})l+(\frac{9}{2}-\frac{18}{\sqrt{5}})\text{.}\hfill \end{array}$$

Fig. 2 and Numerical computation depicted in Table 5 and 6 shows an increasing behavior of eV-degree based indices with the increase in the value of $n,m$ and l. Zhong et al. (2021) developed the quantitative structure–property relationship (QSPR) between physical properties (Docking Score, Binding Affinity, Molecular Weight and Topological Polar Surface) and ve- and eV-degree based topological indices. They examined that among eV-degree based indices, the first eV-degree Zagreb index ( $M^{\mathit{ev}}$ ) predict the molecular weight better than eV-degree Randic index ( $R^{\mathit{ev}}$ ). Hence, the above results of eV-degree based indices are helpful to measure the physical properties of starphene.

Theorem 2

Let H be a structure of starphene, then $${\mathsf{M}}_1^{\alpha \mathit{ve}}\left(H\right)=170m+170n+170l-390$$

Proof 2

The partition of vertex set of starphene based on ve degree of each vertex is depicted in Table 3. By using Table 3, we have first ve-degree Zagreb $\alpha$ -index

$$\begin{array}{cc}\hfill {\mathsf{M}}_1^{\alpha \mathit{ve}}\left(H\right)=& {\displaystyle \sum _{v\in V\left(H\right)}}{\mathrm{\Psi }}_{\mathit{ve}}{\left(v\right)}^2,\hfill \\ \hfill {\mathsf{M}}_1^{\alpha \mathit{ve}}\left(H\right)=& {\left(4\right)}^2\left(6\right)+{\left(5\right)}^2\left(6\right)+{\left(6\right)}^2(2m+2n+2l-12)\hfill \\ \hfill +& {\left(7\right)}^2(2m+2n+2l-12)+{\left(8\right)}^2\left(6\right)\hfill \\ \hfill =& 170m+170n+170l-390\text{.}\hfill \end{array}$$

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Fig. 2

Plots of the indices (a) ${\mathsf{M}}^{\mathit{ev}}$ and (b) ${\mathsf{R}}^{\mathit{ev}}$ .

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Table 3 Vertices partition for ve-degrees of $\mathit{St}[n,m,l]$ .

$\mathbf{\Psi }\left(\mathsf{u}\right)$	${\mathbf{\Psi }}_{\mathsf{ve}}\left(\mathsf{u}\right)$	Frequency
2	4	6
2	5	6
2	6	$2m+2n+2l-12$
3	7	$2m+2n+2l-12$
3	8	6

Fig. 3 shows an increasing behavior of first Zagreb alpha index ( $M_1^{\alpha \mathit{ve}}$ ) with an increase in the values of $n,m$ and l. Süleyman (2017) developed QSPR between physical properties (Enthalpy of vaporization (HVAP), Entropy, Standard enthalpy of vaporization (DHVAP), and Acentric factor) and ve and eV-degree based Zagreb and Randic type indices(Süleyman, 2017). He examines that $M_1^{\alpha \mathit{ve}}$ can be helpful to predict the physical property Acentric Factor. Zhong et al. (2021) analyzed that $M_1^{\alpha \mathit{ve}}$ can predict the molecular weight better than eV-degree Randic index ( $R^{\mathit{ev}}$ ). Hence, the above result of ve-degree based index can be helpful to estimate the physical properties of starphene.

Theorem 3

Let H be a structure of starphene $\mathit{St}[n,m,l]$ structure and $m,n,l⩾3$ , then

(a)   ${\mathsf{M}}_1^{\beta \mathit{ve}}\left(H\right)=66m+66n+66l-144$ .

(b)   ${\mathsf{M}}_2^{\beta \mathit{ve}}\left(H\right)=217m+217n+217l-504$ .

(c)   ${\mathsf{ABC}}^{\mathit{ve}}\left(H\right)=(\frac{4\sqrt{11}}{\sqrt{42}}+\frac{\sqrt{12}}{7})m+(\frac{4\sqrt{11}}{\sqrt{42}}+\frac{\sqrt{12}}{7})n+(\frac{4\sqrt{11}}{\sqrt{42}}+\frac{\sqrt{12}}{7})l$         $+(\frac{3\sqrt{6}}{4}+\frac{6\sqrt{7}}{\sqrt{20}}+\frac{6\sqrt{10}}{\sqrt{35}}-\frac{30\sqrt{11}}{\sqrt{42}}-\frac{6\sqrt{12}}{7}+\frac{3\sqrt{14}}{4}+3)$ .

(d)   ${\mathsf{GA}}^{\mathit{ve}}\left(H\right)=(\frac{8\sqrt{42}}{13}+1)m+(\frac{8\sqrt{42}}{13}+1)n+(\frac{8\sqrt{42}}{13}+1)l+(\frac{8\sqrt{5}}{3}-\frac{60\sqrt{42}}{13}+\frac{24\sqrt{3}}{7}+\sqrt{35}+3)$ .

(e)   ${\mathsf{H}}^{\mathit{ve}}\left(H\right)=\frac{414}{546}m+\frac{414}{546}n+\frac{414}{546}l-\frac{427}{546}$ .

(f)   ${\chi }^{\mathit{ve}}\left(H\right)=(\frac{4}{\sqrt{13}}+\frac{1}{\sqrt{14}})m+(\frac{4}{\sqrt{13}}+\frac{1}{\sqrt{14}})n+(\frac{4}{\sqrt{13}}+\frac{1}{\sqrt{14}})l+(\frac{3}{\sqrt{8}}+\frac{6}{\sqrt{12}}-\frac{30}{\sqrt{13}}+\frac{7}{2})$ .

(g)   ${\mathsf{R}}^{\mathit{ve}}\left(H\right)=(\frac{4}{\sqrt{42}}+\frac{1}{7})m+(\frac{4}{\sqrt{42}}+\frac{1}{7})n+(\frac{4}{\sqrt{42}}+\frac{1}{7})l+(\frac{9}{14}+\frac{6}{\sqrt{20}}+\frac{6}{\sqrt{35}}-\frac{30}{\sqrt{42}}+\frac{6}{\sqrt{48}})$ .

Proof 3

By using the edge partition given in Table 4, we can compute the topological indices based on the ve degree of end vertices of each edge as follows

(a) First Zagreb $\beta$ -index $$\begin{array}{cc}\hfill {\mathsf{M}}_1^{\beta \mathit{ve}}\left(H\right)=& {\displaystyle \sum _{\mathit{uv}\in E\left(H\right)}}\left({\mathrm{\Psi }}_{\mathit{ve}}\right(u)+{\mathrm{\Psi }}_{\mathit{ve}}(v\left)\right),\hfill \\ \hfill {\mathsf{M}}_1^{\beta \mathit{ve}}\left(H\right)=& \left(8\right)\left|E_1^{\ast }\right|+\left(9\right)\left|E_2^{\ast }\right|+\left(12\right)\left|E_3^{\ast }\right|+\left(13\right)\left|E_4^{\ast }\right|+\left(14\right)\left|E_5^{\ast }\right|+\left(14\right)\left|E_6^{\ast }\right|+\left(16\right)\left|E_7^{\ast }\right|\hfill \\ \hfill =& 66m+66n+66l-144\text{.}\hfill \end{array}$$

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Fig. 3

Plot of ${\mathsf{M}}_1^{\alpha \mathit{ve}}$ .

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Table 4 Edges Partition based on ve-degree of end vertices of each edge of $\mathit{St}[n,m,l]$ .

Edge	$\left({\mathbf{\Psi }}_{\mathsf{ve}}\right(\mathsf{u}),{\mathbf{\Psi }}_{\mathsf{ve}}(\mathsf{v}\left)\right)$	Frequency
$E_1^{\ast }$	(4, 4)	3
$E_2^{\ast }$	(4, 5)	6
$E_3^{\ast }$	(5, 7)	6
$E_4^{\ast }$	(6, 7)	$2(2m+2n+2l-15)$
$E_5^{\ast }$	(6, 8)	6
$E_6^{\ast }$	(7, 7)	$m+n+l-6$
$E_7^{\ast }$	(8, 8)	6

Table 5 Numerical results of topological indices for different values of $[n,m,l]$ .

$[n,m,l]$	${\mathsf{M}}^{\mathit{ev}}\left(H\right)$	${\mathsf{R}}^{\mathit{ev}}\left(H\right)$	${\mathsf{M}}_1^{\alpha \mathit{ve}}\left(H\right)$	${\mathsf{M}}_1^{\beta \mathit{ve}}\left(H\right)$	${\mathsf{M}}_2^{\beta \mathit{ve}}(H$ )
$[1,1,1]$	102	3.0415	126	54	147
$[2,2,2]$	510	9.6328	642	252	798
$[3,3,3]$	918	16.2241	1158	450	1449
$[4,4,4]$	1326	22.8154	1674	648	2100
$[5,5,5]$	1734	29.4067	2190	846	2751
$[6,6,6]$	2142	35.9980	2706	1044	3402
$[7,7,7]$	2550	42.5893	3222	1242	4053
$[8,8,8]$	2958	49.1806	3738	1440	4704
$[9,9,9]$	3366	55.7719	4254	1638	5355
$[10,10,10]$	3774	62.3632	4770	1836	6006

(b) Second Zagreb $\beta$ -index $$\begin{array}{cc}\hfill {\mathsf{M}}_2^{\beta \mathit{ve}}\left(H\right)=& {\displaystyle \sum _{\mathit{uv}\in E\left(H\right)}}\left({\mathrm{\Psi }}_{\mathit{ve}}\right(u)\times {\mathrm{\Psi }}_{\mathit{ve}}(v\left)\right),\hfill \\ \hfill {\mathsf{M}}_2^{\beta \mathit{ve}}\left(H\right)=& \left(16\right)\left|E_1^{\ast }\right|+\left(20\right)\left|E_2^{\ast }\right|+\left(35\right)\left|E_3^{\ast }\right|+\left(42\right)\left|E_4^{\ast }\right|+\left(48\right)\left|E_5^{\ast }\right|+\left(49\right)\left|E_6^{\ast }\right|+\left(64\right)\left|E_7^{\ast }\right|\hfill \\ \hfill =& 217m+217n+217l-504\text{.}\hfill \end{array}$$

(c) Atom-bond connectivity index $$\begin{array}{cc}\hfill {\mathsf{ABC}}^{\mathit{ve}}\left(H\right)=& {\displaystyle \sum _{\mathit{uv}\in E\left(H\right)}}\sqrt{\frac{{\mathrm{\Psi }}_{\mathit{ve}}\left(u\right)+{\mathrm{\Psi }}_{\mathit{ve}}\left(v\right)-2}{\left({\mathrm{\Psi }}_{\mathit{ve}}\right(u)\times {\mathrm{\Psi }}_{\mathit{ve}}(v\left)\right)}},\hfill \\ \hfill {\mathsf{ABC}}^{\mathit{ve}}\left(H\right)=& \left(\sqrt{\frac{6}{16}}\right)\left|E_1^{\ast }\right|+\left(\sqrt{\frac{7}{20}}\right)\left|E_2^{\ast }\right|+\left(\sqrt{\frac{10}{35}}\right)\left|E_3^{\ast }\right|+\left(\sqrt{\frac{11}{42}}\right)\left|E_4^{\ast }\right|+\left(\sqrt{\frac{12}{48}}\right)\left|E_5^{\ast }\right|\hfill \\ \hfill +& \left(\sqrt{\frac{12}{49}}\right)\left|E_6^{\ast }\right|+\left(\sqrt{\frac{14}{64}}\right)\left|E_7^{\ast }\right|\hfill \\ \hfill =& (\frac{4\sqrt{11}}{\sqrt{42}}+\frac{\sqrt{12}}{7})(m+n+l)\hfill \\ \hfill +& (\frac{3\sqrt{6}}{4}+\frac{6\sqrt{7}}{\sqrt{20}}+\frac{6\sqrt{10}}{\sqrt{35}}-\frac{30\sqrt{11}}{\sqrt{42}}-\frac{6\sqrt{12}}{7}+\frac{3\sqrt{14}}{4}+3)\text{.}\hfill \end{array}$$

(d)  Geometric-arithmetic index $$\begin{array}{cc}\hfill {\mathsf{GA}}^{\mathit{ve}}\left(H\right)=& {\displaystyle \sum _{\mathit{uv}\in E\left(H\right)}}\frac{2\sqrt{{\mathrm{\Psi }}_{\mathit{ve}}\left(u\right)\times {\mathrm{\Psi }}_{\mathit{ve}}\left(v\right)}}{\left({\mathrm{\Psi }}_{\mathit{ve}}\right(u)+{\mathrm{\Psi }}_{\mathit{ve}}(v\left)\right)},\hfill \\ \hfill {\mathsf{GA}}^{\mathit{ve}}\left(H\right)=& \left(\frac{2\sqrt{16}}{8}\right)\left|E_1^{\ast }\right|+\left(\frac{2\sqrt{20}}{9}\right)\left|E_2^{\ast }\right|+\left(\frac{2\sqrt{35}}{12}\right)\left|E_3^{\ast }\right|+\left(\frac{2\sqrt{42}}{13}\right)\left|E_4^{\ast }\right|+\left(\frac{2\sqrt{48}}{14}\right)\left|E_5^{\ast }\right|\hfill \\ \hfill +& \left(\frac{2\sqrt{49}}{14}\right)\left|E_6^{\ast }\right|+\left(\frac{2\sqrt{64}}{16}\right)\left|E_7^{\ast }\right|\hfill \\ \hfill =& (\frac{8\sqrt{42}}{13}+1)m+(\frac{8\sqrt{42}}{13}+1)n+(\frac{8\sqrt{42}}{13}+1)l+(\frac{8\sqrt{5}}{3}-\frac{60\sqrt{42}}{13}\hfill \\ \hfill +& \frac{24\sqrt{3}}{7}+\sqrt{35}+3)\text{.}\hfill \end{array}$$

(e)  Harmonic index $$\begin{array}{cc}\hfill {\mathsf{H}}^{\mathit{ve}}\left(H\right)=& {\displaystyle \sum _{\mathit{uv}\in E\left(H\right)}}\frac{2}{{\mathrm{\Psi }}_{\mathit{ve}}\left(u\right)+{\mathrm{\Psi }}_{\mathit{ve}}\left(v\right)},\hfill \\ \hfill {\mathsf{H}}^{\mathit{ve}}\left(H\right)=& \left(\frac{2}{8}\right)\left|E_1^{\ast }\right|+\left(\frac{2}{9}\right)\left|E_2^{\ast }\right|+\left(\frac{2}{12}\right)\left|E_3^{\ast }\right|+\left(\frac{2}{13}\right)\left|E_4^{\ast }\right|+\left(\frac{2}{14}\right)\left|E_5^{\ast }\right|+\left(\frac{2}{14}\right)\left|E_6^{\ast }\right|+\left(\frac{2}{16}\right)\left|E_7^{\ast }\right|\hfill \\ \hfill =& \frac{414}{546}m+\frac{414}{546}n+\frac{414}{546}l-\frac{427}{546}\text{.}\hfill \end{array}$$

(f)  Sum-connectivity index $$\begin{array}{cc}\hfill {\chi }^{\mathit{ve}}\left(H\right)=& {\displaystyle \sum _{\mathit{uv}\in E\left(H\right)}}{\left({\mathrm{\Psi }}_{\mathit{ve}}\right(u)+{\mathrm{\Psi }}_{\mathit{ve}}(v\left)\right)}^{-\frac{1}{2}},\hfill \\ \hfill {\chi }^{\mathit{ve}}\left(H\right)=& {\left(8\right)}^{-\frac{1}{2}}\left|E_1^{\ast }\right|+{\left(9\right)}^{-\frac{1}{2}}\left|E_2^{\ast }\right|+{\left(12\right)}^{-\frac{1}{2}}\left|E_3^{\ast }\right|+{\left(13\right)}^{-\frac{1}{2}}\left|E_4^{\ast }\right|+{\left(14\right)}^{-\frac{1}{2}}\left|E_5^{\ast }\right|\hfill \\ \hfill +& {\left(14\right)}^{-\frac{1}{2}}\left|E_6^{\ast }\right|+{\left(16\right)}^{-\frac{1}{2}}\left|E_7^{\ast }\right|\hfill \\ \hfill =& (\frac{4}{\sqrt{13}}+\frac{1}{\sqrt{14}})m+(\frac{4}{\sqrt{13}}+\frac{1}{\sqrt{14}})n+(\frac{4}{\sqrt{13}}+\frac{1}{\sqrt{14}})l+(\frac{3}{\sqrt{8}}+\frac{6}{\sqrt{12}}-\frac{30}{\sqrt{13}}+\frac{7}{2})\text{.}\hfill \end{array}$$

(g)  Randic index $$\begin{array}{cc}\hfill {\mathsf{R}}^{\mathit{ve}}\left(H\right)=& {\displaystyle \sum _{\mathit{uv}\in E\left(H\right)}}{\left({\mathrm{\Psi }}_{\mathit{ve}}\right(u)\times {\mathrm{\Psi }}_{\mathit{ve}}(v\left)\right)}^{-\frac{1}{2}},\hfill \\ \hfill {\mathsf{R}}^{\mathit{ve}}\left(H\right)=& {\left(16\right)}^{-\frac{1}{2}}\left|E_1^{\ast }\right|+{\left(20\right)}^{-\frac{1}{2}}\left|E_2^{\ast }\right|+{\left(35\right)}^{-\frac{1}{2}}\left|E_3^{\ast }\right|+{\left(42\right)}^{-\frac{1}{2}}\left|E_4^{\ast }\right|+{\left(48\right)}^{-\frac{1}{2}}\left|E_5^{\ast }\right|\hfill \\ \hfill +& {\left(49\right)}^{-\frac{1}{2}}\left|E_6^{\ast }\right|+{\left(64\right)}^{-\frac{1}{2}}\left|E_7^{\ast }\right|\hfill \\ \hfill =& (\frac{4}{\sqrt{42}}+\frac{1}{7})m+(\frac{4}{\sqrt{42}}+\frac{1}{7})n+(\frac{4}{\sqrt{42}}+\frac{1}{7})l+(\frac{9}{14}+\frac{6}{\sqrt{20}}+\frac{6}{\sqrt{35}}-\frac{30}{\sqrt{42}}+\frac{6}{\sqrt{48}})\text{.}\hfill \end{array}$$

Figs. 4–7 shows the behavior of the computed topological indices based on ve degree. The values of all these topological indices increases with the increase in the value of $m,n$ and l. Süleyman (2017) examines that the second Zagreb beta index can be used to predicts the entropy and the Randic index can be helpful to predicts the Enthalpy of vaporization and Standard enthalpy of vaporization. Zhong et al. (2021) investigated that the first Zagreb beta index ( $M_1^{\beta \mathit{ve}}$ ) predicts the molecular weight and Topological Polar Surface better than other ve-degree based topological indices. Overall, $M_1^{\beta \mathit{ve}}$ is the best predictor of the molecular weight and Topological Polar Surface in all ve- and eV-degree based indices. Hence, the above results are helpful to measure the physical properties of starphene.

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Fig. 4

plots of the indices (a) ${\mathsf{M}}_1^{\beta \mathit{ve}}$ and (b) ${\mathsf{M}}_2^{\beta \mathit{ve}}$ .

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Fig. 5

plots of the indices (a) ${\mathsf{ABC}}^{\mathit{ve}}$ and (b) ${\mathsf{GA}}^{\mathit{ve}}$ .

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Fig. 6

Plots of the indices (a) ${\mathsf{H}}^{\mathit{ve}}$ and (b) ${\chi }^{\mathit{ve}}$ .

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Fig. 7

Plot of the indices ${\mathsf{R}}^{\mathit{ve}}$ .

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3 Structure of fenestrene $F[n,m]$

In this section, we discuss about the fenestrene structure which belongs to the class of benzenoid systems. Acenes are benzene rings fused linearly. We denote the structure of fenestrene by $F[n,m]$ , where m and n denoted the number of hexagons in each row and column respectively. The molecular structure of fenestrene is depicted in Fig. 8. The structure of $F[n,m]$ has $8(m+n)$ vertices. For $n=1,2$ the graph of $F[n,m]$ has $6\mathit{mn}+7m+5n+1$ edges and for $n⩾3$ it has $20m+10n-10$ edges. Let ${\epsilon }_{(i,j)}$ denotes the edge set containing the edges of $F[n,m]$ with end vertices of degree i and j respectively. The edge set of $F[n,m]$ can be partitioned based on the degree of end vertices as follows: ${\epsilon }_{(2,2)}$ containing $4m+2$ edge, ${\epsilon }_{(2,3)}$ containing $8m+8n-12$ edges and ${\epsilon }_{(3,3)}$ containing $8m+2n$ edges.

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Fig. 8

Fenestrene structure $F[n,m]$ .

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