Group Theoretic Approach towards the Balaban Index of Catacondensed Benzenoid Systems and Linear Chain of Anthracene

Abstract

In this work, we present the analytical closed forms of the Balaban index for anthracene and catacondensed benzenoid systems using group theoretic techniques. The Balaban index is a distance-based topological index that provides valuable information about the properties of chemical structures. We emphasize the importance of determining analytical closed forms of the Balaban index for catacondensed benzenoid systems and linear chains of anthracene, as it enables a deeper understanding of these systems and their behavior. Our analysis utilizes the group action of the automorphism group of these chains on the set of vertices, which refer to the points where the chains intersect. In future work, we plan to determine the Balaban index of other polymeric linear chains using group theoretic techniques and extend the applications of this index to other fields, such as materials science and biology. It is clear that the Balaban index remains a valuable tool in theoretical and computational chemistry, and its applications are constantly evolving.

1. Introduction

The pharmaceutical industry continuously develops new medicinal drugs to combat resistance in pathogenic agents and cope with newly discovered infections, such as mutated bacteria, viral infections (e.g., Ebola virus, HIV), and prions. A key challenge in drug development is predicting the physical, chemical, and biological properties of organic compounds based on their molecular structure. Topological indices, which are mathematical values used to investigate various physicochemical properties of molecule structures, provide a cost-effective and time-efficient method for reducing laboratory experiments [1,2]. The Balaban index, developed in the early 1980s by Alexandru T. Balaban, is a prominent topological index. It offers an ordering similar to the Wiener index but with lower degeneracy and a higher discriminating ability, making it particularly useful for larger or more complex molecules [3]. The issue of degeneracy arises when topological indices fail to distinguish between molecules with the same number of atoms or cycles. The Balaban index mitigates this problem better than other indices, proving valuable in drug design and discovery due to its low degeneracy and high discriminatory power. Mathematically, the Balaban index J for a structural graph $C_k$ is defined as follows:

$$\begin{array}{c}\hfill J\left(C_k\right)={\displaystyle \frac{p}{\mu +1}}\sum _{e=v_1{v}_2}{\left[d\left(v_1\right)d\left(v_2\right)\right]}^{-1/2}\end{array}$$

where $\mu =p-q+1$ is referred to as the cyclomatic number of graph $C_k$, d represents the distance between two vertices, and p and q are the number of links and nodes respectively.

The Balaban index is particularly effective for complex structures with numerous bonds and heteroatoms. In QSAR/QSPR studies, especially those involving alkanes, the Balaban index has been associated with properties like boiling points, density, refractive index, and octane values [4,5,6]. It has also been applied in drug development and the study of the complexity of super-molecular substituted sulfonamides [7,8,9,10,11,12,13,14]. Furthermore, the index has been used to determine the properties of polymers and predict the melting and glass transition temperatures of linear macromolecules [15,16,17,18]. In various biological studies, including those optimizing physiologically active lead compounds, the Balaban index has been employed alongside other chemical descriptors [19,20,21,22,23].

Anthracene, discovered in 1832, and its derivatives, such as anthracene glycosides, have been used for academic, industrial, and medical purposes [24,25,26]. This paper aims to compute the Balaban index for anthracene and the catacondensed benzenoid system to enhance understanding of their molecular properties and structural characteristics in theoretical and polymeric chemistry [27,28].

Catacondensed benzenoids, which lack internal vertices (carbon atoms bonded by three hexagonal rings), are isomeric with the formula $C_{4r+2}{H}_{2r+4}$,where variable r represent the number of hexagonal rings in the benzenoid structure. These compounds have a long research history, with foundational insights from chemists like Kekulé and Fries [29,30,31,32,33,34,35,36]. We focus on linear benzenoids, the simplest type of unbranched benzenoids, and discuss their physicochemical properties as documented in the literature [37,38,39,40].

In this article, we employ group-theoretic methods, using the automorphism group and group action of graphs, to systematically generate indices for linear chains and catacondensed benzenoid systems. This approach leverages structural symmetries to reduce the number of calculations and avoid redundancy. The use of group theory in computational chemistry provides a powerful and efficient tool for generating topological indices. We apply these methods to compute the Balaban index for long linear chains of anthracene and catacondensed benzenoid systems, organic structures with significant physicochemical properties. The computation of the automorphism group for these structures represents a novel contribution to the field [41]. Additionally, we calculate the Balaban index for various dendrimers and compounds, further demonstrating the method’s applicability in different chemical contexts. By employing the Balaban index and other topological indices, researchers can gain a better understanding of the physical and chemical properties of compounds, facilitating the design of new drugs and materials [42,43,44,45,46,47]. Further relevant research can be found in [48,49,50,51,52].

2. Result and Discussion

Moving toward our objectives, we must first obtain the following outcome before moving on to the primary proof for anthracene polymer. Anthracene is represented by the notation $A_l$, where l first stands for the odd and then the even number of anthracene units in the chain, and r stands for the number of subunit steps (which is always less than L). First, we arrive at a crucial conclusion that will be applied to the main proof.

Proposition 1.

For $A_{l}wherel\in \{l:l=2w-1wherew\in Z^{+}\}$

$$\begin{array}{c}\hfill d\left(v_s\right)=\left\{\begin{array}{c}98r^2-84r+\alpha \left(s\right)\mathit{if}s+r\mathit{is}\mathit{even}\hfill \\ 98r^2-84r+\beta \left(s\right)\mathit{if}s+r\mathit{is}\mathit{odd}\hfill \end{array}\right.\end{array}$$

where $\alpha \left(s\right)=21-4s+2s^2$ and $\beta \left(s\right)=19-4s+2s^2$.

Proof. 

We use it on a case-by-case basis.

Case 1: On s = 1 we will employ the inductive method. As we can see,

$$\begin{array}{c}\hfill d\left(v_1\right)=19-84r+98r^2\end{array}$$

Step 1: let $r=1$

$$\begin{array}{c}\hfill d\left(v_1\right)=33\end{array}$$

For $r=1$, this is true.

Step 2: Now, we suppose it is true for $r=n$, i.e., $d\left(v_1\right)=19-84n+98n^2$

We will prove it is true for $r=n+1$

$$\begin{array}{cc}\hfill d\left(v_1\right)=& 2\times \left(\right(7n-3)+14n+2\times ((7n-2)+(7n-1)+(7n+1)+(7n+2)\hfill \\ & +(7n+3))+(7n+4))19-84n+98n^2\hfill \end{array}$$

it will become,

$$\begin{array}{ccc}\hfill d\left(v_1\right)& =& 98+196n+98n^2-84-84n+19\hfill \\ & =& 98{(1+n)}^2-84(1+n)+19\hfill \end{array}$$

Thus, the proof has been completed. The same is valid for every even $s+r$.

Case 2: we will employ the inductive method on $s=2$.

$$\begin{array}{c}\hfill d\left(v_2\right)=98r^2-84r+19\end{array}$$

Step 1: Let $r=1$

$$\begin{array}{c}\hfill d\left(v_2\right)=33\end{array}$$

For $r=1$, it is true.

Step 2: Now we suppose it is true for $r=n$. i.e., $d\left(v_2\right)=98n^2-84n+19$

We will prove it is valid for $r=n+1$

$$\begin{array}{cc}\hfill d\left(v_2\right)=& 2\times \left(\right(7n-3)+2\times \{(7n-2)+(7n-1)\}+14n+2((7n+1)+(7n+2)\hfill \\ & +(7n+3))+(7n+4))+98n^2-84n+19\hfill \end{array}$$

then

$$\begin{array}{ccc}\hfill d\left(v_2\right)& =& \hfill 98n^2+196n+98-84n-65\hfill \\ & =& \hfill 98{(n+1)}^2-84(n+1)+19\hfill \end{array}$$

Hence, the proof has been completed. Similarly, its validity holds for all odd $s+r$. □

We will now derive the Balaban index of $A_l$ by using the above Proposition 1.

Theorem 1.

Let J denote the B alaban index and $A_{l}wherel\in \{l:l=2w-1wherew\in Z^{+}\}$ denote the linear chain of anthracene, then

$$\begin{array}{cc}\hfill J\left(A_l\right)=& {\displaystyle \frac{18l-2}{4l}}[\sum _{s=1}^{\frac{7l-1}{2}}\sqrt{(98r^2-84r+\alpha \left(s\right))(98r^2-84r+\beta \left(s\right))}\hfill \\ & +\sum _{s=1}^{\frac{1+7l}{2}}{\displaystyle \frac{1}{d\left(v_s\right)}}-\sum _{s=1}^{\frac{1+l}{2}}{\displaystyle \frac{1}{d\left(v_{7s-6}\right)}}-\sum _{s=1}^{\frac{1+l}{2}}{\displaystyle \frac{1}{d\left(v_{7s-4}\right)}}-\sum _{s=1}^{\frac{1+l}{2}}{\displaystyle \frac{1}{d\left(v_{7s-1}\right)}}]\hfill \end{array}$$

Proof. 

Linear chain $A_{l}wherel\in \{l:l=2w-1wherew\in Z^{+}\}$ has “two” two-fold symmetry axes. We study both of the symmetry axes; one is vertical while the other is horizontal in the plane. The middle point of the edge, which is in the central edge of the anthracene, is the center of both symmetry axes. The horizontal axis cuts across a group of edges, whereas the vertical axis runs parallel to a central edge in anthracene. Suppose $\varphi \in Aut\left(A_m\right)$, then every carbon atom x and $\varphi \left(x\right)$; either it is bridging carbon or not, with lye in the same step $i(1\le i\le \frac{7l+1}{2})$. The argument is that the eccentricity of both x and $\varphi \left(x\right)$ is equal. This implies $Aut\left(N_l\right)=\{{\varphi }_0,{\varphi }_1,{\varphi }_2,{\varphi }_3\}$ for all $l\in \{l:l=2w-1wherew\in Z^{+}\}$.

In the structural graph drawn in Figure 1, three anthracene rings make up the polymeric linear chain system, which is a benzenoid derivative. Bridging carbon atoms connect three cyclobutadiene rings to form an anthracene ring, and there are 20 bridging carbon atoms.

The polymeric linear chain system, which is a benzenoid derivative, is made up of three anthracene rings in the structural graph described in Figure 2. There are 20 bridging carbon atoms that join three cyclobutadiene rings to make a anthracene ring.

The structural graph defined in Figure 3 has three anthracene rings. The symmetry axis is horizontal.

The symmetry axis is vertical in the structure defined in Figure 4 of three anthracene rings.

In Figure 5, the symmetry axis is the last automorphism of the structure of three anthracene rings, and it is composed of both consecutive symmetry axes.

In Figure 6, multitudinous anthracene rings make up the polymeric linear chain system, which is a benzenoid derivative.

Let $E_s$ be a represented by $x_s{x}_{s+1}$ then $p_l=\left|E_l\right|=18l-2$ and $q_l=\left|V_l\right|=14l$, to calculate the index of $A_l$, we found $\mu \left(A_l\right)=4l-1$.

When $Aut\left(A_l\right)$ acts on $V\left(A_l\right)$, then the orbits of $Aut\left(A_l\right)$ on $V\left(A_l\right)$ are $V_1=\{x_4,x_1\}$, $V_2=\{x_6,x_5,x_3,x_2\},$…,$V_{\frac{7l+1}{2}}=\{x_{7l+2},x_{7l+3},x_{7i+2},x_{7i+3}\}$. Let $v_1=x_1$ and $v_s\in V_n,$ where $1\le n\le \frac{7l+1}{2}$.

The orbits of the action when $Aut\left(A_l\right)$ acts on $E\left(A_l\right)$ are as follows:

$$\begin{array}{ccc}\hfill p_1=\left|E_1\right|& =& \{x_2{x}_1,x_4{x}_3,x_5{x}_4,x_1{x}_6\},\hfill \\ \hfill p_2=\left|E_1\right|& =& \{x_3{x}_2,x_6{x}_5\},\hfill \\ \hfill p_3=\left|E_2\right|& =& \{x_7{x}_2,x_{10}{x}_3,x_{14}{x}_5,x_{11}{x}_6\},\hfill \\ \hfill p_4=\left|E_3\right|& =& \{x_8{x}_7,x_{10}{x}_9,x_{14}{x}_{13},x_{12}{x}_{11}\},\hfill \\ \hfill \dots \dots \dots \dots \\ \hfill |E_{\frac{11l-1}{2}}|=p_{\frac{11l-1}{2}}& =& \{x_{7l+2}{x}_{7l+3},x_{7i+2}{x}_{7i+3}\}.\hfill \end{array}$$

Thus,

$$\begin{array}{c}\hfill J\left(A_l\right)={\displaystyle \frac{18l-1}{4l-1+1}}[\sum _{s=1}^{\frac{7l-1}{2}}{\displaystyle \frac{4}{\sqrt{d\left(v_s\right).d\left(v_{s+1}\right)}}}+\sum _{s=1}^{\frac{7l+1}{2}}{\displaystyle \frac{1}{d\left(v_s\right)}}-\sum _{s=1}^{\frac{l+1}{2}}{\displaystyle \frac{1}{d\left(v_{7s-6}\right)}}-\sum _{s=1}^{\frac{l+1}{2}}{\displaystyle \frac{1}{d\left(v_{7s-4}\right)}}-\sum _{s=1}^{\frac{l+1}{2}}{\displaystyle \frac{1}{d\left(v_{7s-1}\right)}}]\end{array}$$

and where

$$\begin{array}{c}\hfill \sum _{s=1}^{\frac{7l+1}{2}}{\displaystyle \frac{1}{d\left(v_s\right)}}={\displaystyle \frac{1}{d\left(v_1\right)}}+{\displaystyle \frac{1}{d\left(v_2\right)}}+....+{\displaystyle \frac{1}{d(v_{\frac{7l+1}{2}})}},\end{array}$$

and

$$\begin{array}{c}\hfill \sum _{s=1}^{\frac{l+1}{2}}{\displaystyle \frac{1}{d\left(v_{7s-6}\right)}}={\displaystyle \frac{1}{d\left(v_1\right)}}+{\displaystyle \frac{1}{d\left(v_8\right)}}+....+{\displaystyle \frac{1}{d(v_{\frac{l+1}{2}})}}.\end{array}$$

By using Proposition 1,

$$\begin{array}{cc}\hfill J\left(A_l\right)=& {\displaystyle \frac{18l-2}{4l}}[\sum _{s=1}^{\frac{7l-1}{2}}\sqrt{(98r^2-84r+\alpha \left(s\right))(98r^2-84r+\beta \left(s\right))}\hfill \\ & +\sum _{s=1}^{\frac{7l+1}{2}}{\displaystyle \frac{1}{d\left(v_s\right)}}-\sum _{s=1}^{\frac{l+1}{2}}{\displaystyle \frac{1}{d\left(v_{7s-6}\right)}}-\sum _{s=1}^{\frac{l+1}{2}}{\displaystyle \frac{1}{d\left(v_{7s-4}\right)}}-\sum _{s=1}^{\frac{l+1}{2}}{\displaystyle \frac{1}{d\left(v_{7s-1}\right)}}]\hfill \end{array}$$

In

Table 1. The Balaban index of $J\left(A_l\right),1\le l\le 999$.

l	J
1.0	1.68384
5.0	1.1285
9.0	1.0211
11	1.01312
15	0.088484
19	0.87654
99	0.00612570
999	0.000193983

, the Balaban index $J\left(A_l\right)$ is computed, for various odd anthracene structures.

We now proceed to our next finding, a polymeric linear chain of anthracene, once more represented by the symbol $A_l$, where $l\in \{l:l=2w\mathrm{where}w\in Z^{+}\}$. First, we write an important result that is driven, like the previous proposition, to be used in the main proof. Its proof is similar to that of Proposition 1. □

Proposition 2.

For $A_{l}wherel\in \{l:l=2wwherew\in Z^{+}\}$

$$\begin{array}{c}\hfill d\left(v_s\right)=\left\{\begin{array}{c}98r^2+14r+\alpha \left(s\right)\mathit{if}s+r\mathit{is}\mathit{even}\hfill \\ 98r^2+14r+\beta \left(s\right)\mathit{if}s+r\mathit{is}\mathit{odd}\hfill \end{array}\right.\end{array}$$

where $\alpha \left(s\right)=2s^2-2s$ and $\beta \left(s\right)=2s^2-2s+2$.

We have used the above proposition to obtain the following result.

Theorem 2.

Let J denote the Balaban index while $A_{l}wherel\in \{l:l=2wwherew\in Z^{+}\}$ denotes the linear chain of anthracene, then

$$\begin{array}{ccc}\hfill J\left(A_l\right)=& & \hfill {\displaystyle \frac{18l-2}{4l}}[\sum _{s=1}^{\frac{7l-2}{2}}\sqrt{(98r^2+14r+\alpha \left(s\right))(98r^2+14r+\beta \left(s\right))}\hfill \\ & & \hfill +\sum _{s=1}^{\frac{7l}{2}}{\displaystyle \frac{1}{d\left(v_s\right)}}-\sum _{s=1}^{\frac{l}{2}}{\displaystyle \frac{1}{d\left(v_{7s-5}\right)}}-\sum _{s=1}^{\frac{l}{2}}{\displaystyle \frac{1}{d\left(v_{7s-3}\right)}}-\sum _{s=1}^{\frac{l}{2}}{\displaystyle \frac{1}{d\left(v_{7s-1}\right)}}]\hfill \end{array}$$

Proof of the above theorem is similar to the proof of Theorem 1. Due to proof similarity, instead of proving it, we directly compute the Balaban index of $A_l$. In

Table 2. The Balaban index of of $J\left(A_l\right),2\le l\le 1000$.

l	J
2	1.4790
6	1.3797
10	1.1076
14	1.0974
18	0.9269
100	0.030136
1000	0.000268172

, the Balaban index $J\left(A_l\right)$ is computed, for several even anthracene structures.

We are now moving toward our outcome, a catacondensed benzenoid system. The catacondensed benzenoid system is represented by the notation $O_l$, where l is the parameter representing the number of subunits in the system and r is the number of steps; in this system, l and r are always equal.

Proposition 3.

For $O_l$

$$\begin{array}{c}\hfill d\left(v_s\right)=\left\{\begin{array}{c}\alpha \left(s\right)+(4s-4)r+12r^2\mathit{if}s\mathit{is}\mathit{even}\hfill \\ \beta \left(s\right)+(4s-4)r+12r^2\mathit{if}s\mathit{is}\mathit{odd}\hfill \end{array}\right.\end{array}$$

where $\alpha \left(s\right)=2{(s-1)}^2-1$ and $\beta \left(s\right)=2{(s-1)}^2-3$.

We now calculate the Balaban index of $O_l$ with the help of the above Proposition 3.

Theorem 3.

Let the Balaban index be denoted by J while the catacondensed benzenoid system is denoted by $O_{l}wherel\in Z^{+}$, then

$$\begin{array}{c}J\left(O_l\right)=[\sum _{s=1}^{\frac{2l+1}{3}}{\displaystyle \frac{6}{\sqrt{16l^2[9l^2+12l-2(7-24{(s-1)}^2)]+4{(s-1)}^2[4(2l-3)+{(s-1)}^2]-192l+143}}}\\ +{\displaystyle \frac{3}{d\left(v_1\right)}}+\sum _{s=2}^{\frac{2l+1}{3}}{\displaystyle \frac{3}{d\left(v_{2s-1}\right)}}]\frac{15l-9}{3l-1}\end{array}$$

Proof. 

The system $O_l$ has “three” two-fold symmetry axes. $O_l$ has a benzene in the center, which is also the center of all symmetry axes in the plane. Each symmetry axis cuts through a number of groups of edges. Suppose $\psi \in Aut\left(O_l\right)$, then every carbon atom x and $\psi \left(x\right)$, whether it is bridging carbon or not, will appear in step $s(1\le s\le \frac{2l+1}{3})$. The reason for this is that the eccentricity of both x and $\psi \left(x\right)$ is equal. This implies $Aut\left(O_l\right)=\{{\psi }_0,{\psi }_1,{\psi }_2,{\psi }_3,{\psi }_4,{\psi }_5\}$ for all $l\in Z^{+}$.

Let $E_s$ be represented by $x_s{x}_{s+1}$, then $|E_l|=15l-9$ and $|V_l|=12l-6$. To derive the Balaban index of $C_l$, we derive that $\mu \left(O_l\right)=3l-2$. In the graph drawn in Figure 7, many benzene rings comprise the polymeric linear chains to make up a catacondensed benzinoid system. Polymeric linear chains are a type of homocyclic conjugated molecule with a long chain of subunits. The major moiety in their structures is the Benzinoid system. Furthermore, they are capable of transporting carbon atoms that are shared by the vertices of two or more rings and there are 12 bridging carbon atoms in the structural graph.

In the structural graph described in Figure 8 of $O_3$, the symmetry axis is horizontal,

In the structural graph of the $O_3$ catacondensed benzinoid system drawn in Figure 9, the axis of symmetry is tilt.

The structural graph, defined in Figure 10 of the $O_3$ catacondensed benzinoid system, which, we obtained from identity automorphism, also shows a slightly tilted symmetry axis.

In the graph defined in Figure 11, the symmetry is the automorphism of the structure of $O_3$, and it is composed of the two first symmetry axes.

The structural graph of $O_3$ described in Figure 12 is also obtained by the composition of two automorphisms. These automorphisms are used to define their orbits.

A type of homocyclic conjugated molecule with several subunits arranged in chains are known as catacondensed benzenoid complexes. The primary moiety in their structures is the benzenoid system. Additionally, they carry carbon atoms that are shared by the atoms of two or more rings. The catacondensed benzenoid system in the Figure 13 structural graph is made up of infinite benzene rings.

When $Aut\left(O_l\right)$ acts on $V\left(O_l\right)$ then orbits of $Aut\left(O_l\right)$ on $V\left(O_l\right)$ are $V_1=\{x_2,x_1,x_4,x_3,x_6,x_5\}$, $V_2=\{x_{10},x_7,x_{15},x_{18},x_{14},x_{11}\},$…,$V_{2l-1}=\{x_{4l+2},x_{4l+3},x_{4i+2},x_{4i+3},x_{4g+2},x_{4g+3}\}$. Let $x_1=v_1$ and $v_s\in V_n,$ where $1\le n\le 2l-1$.

$$\begin{array}{c}\hfill J\left(O_l\right)={\displaystyle \frac{p}{\mu +1}}\sum _{x_2{x}_1\in E\left(O_l\right)}{\displaystyle \frac{1}{\sqrt{d\left(x_2\right)d\left(x_1\right)}}}\end{array}$$

The orbits of $Aut\left(O_l\right)$ when acting on $E\left(O_l\right)$ are as follows:

$$\begin{array}{ccc}\hfill |E_1| =p_1& =& \{x_6{x}_5,x_1{x}_6,x_2{x}_1,x_3{x}_2,x_4{x}_3,x_5{x}_4\},\hfill \\ \hfill |E_1| =p_2& =& \{x_7{x}_2,x_{10}{x}_3,x_{18}{x}_4,x_{15}{x}_5,x_{14}{x}_6,x_{11}{x}_1\},\hfill \\ \hfill |E_2| =p_3& =& \{x_8{x}_7,x_{10}{x}_9,x_{16}{x}_{15},x_{18}{x}_{17},x_{12}{x}_{11},x_{14}{x}_{13}\},\hfill \\ \hfill |E_3| =p_4& =& \{x_9{x}_8,x_{17}{x}_{16},x_{13}{x}_{12}\},\hfill \\ \hfill \dots \dots \dots \dots \\ \hfill |E_{2l}| =p_{2l}& =& \{x_{2l-1}{x}_{2l},x_{2i-1}{x}_{2i},x_{2g-1}{x}_{2g}\}.\hfill \end{array}$$

then

$$\begin{array}{c}\hfill J\left(O_l\right)={\displaystyle \frac{15l-1}{3l-2+1}}[\sum _{i=1}^{2l-1}{\displaystyle \frac{6}{\sqrt{d\left(v_s\right).d\left(v_{s+1}\right)}}}+{\displaystyle \frac{3}{d\left(v_1\right)}}+\sum _{s=2}^{2l-1}{\displaystyle \frac{3}{d\left(v_{2s-1}\right)}}]\end{array}$$

where

$$\begin{array}{c}\hfill \sum _{s=2}^{2l-1}{\displaystyle \frac{3}{d\left(v_{2s-1}\right)}}={\displaystyle \frac{3}{d\left(v_3\right)}}+....+{\displaystyle \frac{3}{d\left(v_{2l-1}\right)}}.\end{array}$$

with the help of above Proposition 3,

$$\begin{array}{c}J\left(O_l\right)=[\sum _{s=1}^{2l-1}{\displaystyle \frac{6}{\sqrt{16l^2[9l^2+12l-2(7-24{(s-1)}^2)]+4{(s-1)}^2[4(2l-3)+{(s-1)}^2]-192l+143}}}\\ +{\displaystyle \frac{3}{d\left(v_1\right)}}+\sum _{s=2}^{2l-1}{\displaystyle \frac{3}{d\left(v_{2s-1}\right)}}]\frac{15l-9}{3l-1}\end{array}$$

In

Table 3. The Balaban index of of $J\left(O_l\right),1\le l\le 1000$.

l	J
1	2
5	1.4689
12	1.1149
15	0.9952
40	0.9103
80	0.0531
200	0.00823
500	0.0004317

, the Balaban index $J\left(O_l\right)$ is computed, for several Catacondensed benzenoid structures.

□

3. Conclusions

Catacondensed benzenoid Systems and linear chains of anthracene have infinite uses. To make use of the symmetries contained in these polymeric structures, we applied group theory tools, specifically group action and automorphism groups, in the current research. To reduce and minimize the trauma of computation, we computed Balaban indices for these chains using these theoretical ideas from classics of group theory.

This approach not only streamlines the computation process but also provided deeper insights into the symmetrical properties of these molecules. The application of group theory allowed for a systematic analysis, enhancing our understanding of their structural characteristics. The Balaban indices, which incorporate aspects of graph connectivity and symmetry, proved effective in characterizing these systems. Our research underscores the value of integrating mathematical frameworks with chemical analysis, paving the way for further advancements in materials science and molecular engineering. Moreover, applications of the proposed method can be seen in the domain of electromagnetic fields [53], enhanced channel estimation [54] and multiplexing communication systems [55] in near future.