Entropies and Degree-Based Topological Indices of Coronene Fractal Structures

Abstract

Molecular fractals are geometric patterns that appear self-similar across all length scales and are constructed by repeating a single unit on a regular basis. Entropy, as a core thermodynamic function, is an extension based on information theory (such as Shannon entropy) and is used to describe the topological structural complexity or degree of disorder in networks. A topological index is a numeric quantity associated with a network or a graph that characterizes its whole structural properties. In this study, we focus on fractal structures formed by systematically repeating a fixed unit of coronene, a polycyclic aromatic hydrocarbon composed of six benzene rings fused in a hexagonal pattern. In this paper, three types of coronal fractal structures, namely zigzag (ZHCF), armchair (AHCF), and rectangular (RCF), are studied, and their five degree-based topological indices and corresponding entropies are calculated.

1. Introduction

In the field of geometric studies, fractal structures have emerged as a focal point due to their self-similar patterns, which are created through the iterative attachment of fixed units in a deterministic or random manner.manner. A classic example of a deterministic fractal structure is the Sierpiński gasket [1].

These fractals were initially introduced to delineate the irregular shapes found in nature [2], and since then, various architectures have been devised to highlight their significance and mimic real-world phenomena [3]. The application of fractals extends across multiple domains of science and technology, including computer graphics, fractal neural networks, and engineering, owing to their resemblance to numerous physical structures [4,5,6,7]. Notably, the discovery of fractal molecular structures in the field of molecular chemistry has led to extensive research on these fractals within a molecular context. Among the earliest chemical structures used for modeling, generating, and analyzing fractal molecules were the benzenoid systems, which encompass a vast array of compounds and have a rich research base [8,9,10,11,12,13,14].

Coronene exemplifies a highly symmetrical and planar benzenoid compound, comprising seven peri-fused aromatic benzene rings. Coronene and pyrene belong to an extensive family of over 100 polycyclic aromatic hydrocarbons (PAHs), which are formed during the incomplete combustion of organic substances, such as petroleum and fossil fuels. While coronene is a significant member of this family, it is produced in smaller quantities than pyrene, and certain other PAHs may present greater risks due to their condensed nature and chemical properties. Klein et al. [1] conducted an investigation into the family of deterministic benzenoid fractals. They systematically repeated benzene units at different stages, with the first stage being benzene and the second stage comprising benzenoids such as kekulene, coronene, and singly connected benzene. Specifically, the coronoid family of benzenoid fractals is generated by synthesizing six benzenes into coronene, and then fusing six coronenes to produce the third stage, as illustrated in Figure 1. In general, when $n>3$, six $(n-1)$ th-stage fractal structures can be synthesized to form an nth-stage fractal of the coronoid family. Plavsić et al. [15] investigated the aromatic properties of these fractal benzenoids based on Clar structures. El-Basil [16] has established the golden mean value ($\tau =1.618033989$) as the characteristic scaling factor for benzenoid fractals. Despite the fact that the synthesis possibilities of these benzenoid fractals were discussed in the 1990s, their synthesis remains a challenging task. The objective of this study is to provide a theoretical characterization of these coronene-based fractals in order to gain a better understanding of their behavior and properties.

All graphs and networks considered in this study are simple, undirected graphs. Let $V\left(G\right)$ and $E\left(G\right)$ represent the vertex set and edge set of network G, respectively. The degree of a vertex, i, is the number of edges incident to i, denoted by $d_i$. The standard notations are mainly followed in [17]. A topological index is a numerical descriptor derived from the structural graph of molecular compounds, providing insights into their chemical and physical properties, as well as their biological activity [18,19,20].

The reciprocal Randić index [21] is defined as

$$RR\left(G\right)=\sum _{uv\in E\left(G\right)}\sqrt{d_u{d}_v}.$$

Shigehalli et al. [22] demonstrated that the reciprocal Randić index exhibits high correlation with entropy, mean radius, and the acentric factor of octane isomers.

The reduced second Zagreb index [23] is given by

$$R{M}_2(G)=\sum _{uv\in E\left(G\right)}(d_u-1)(d_v-1).$$

Mahboob et al. [24] established that the reduced second Zagreb index is effective for analyzing hepatitis medicine.

The Balaban index [25,26] is defined as

$$J(G)=\sum _{uv\in E\left(G\right)}\left({\displaystyle \frac{m}{m-n+2}}\right){\displaystyle \frac{1}{\sqrt{d_u{d}_v}}},$$

where $n=\left|V\right(G\left)\right|$ and $m=\left|E\right(G\left)\right|$. Thakur et al. [27] found that the Balaban index is useful for modeling the carbonic anhydrase inhibitory activity of the sulfonamides.

Based on Zagreb indices, Azari et al. [28] introduced an another type of topological index called the generalized Zagreb index or the $(a,b)$-Zagreb index in 2011, and it is defined as

$$Z_{a,b}(G)=\sum _{uv\in E\left(G\right)}\left(d_u^a{d}_v^b+d_u^b{d}_v^a\right).$$

In 2024, Gutman [29] introduced the elliptic sombor index:

$$ESO(G)=\sum _{uv\in E\left(G\right)}(d_u+d_v)\sqrt{d_u^2+d_v^2}.$$

Shannon [30] first introduced the concept of entropy in his 1948 work. Entropy is the quantity of thermal energy per unit temperature in a system that is not unavailable for useful work [31,32]. This concept is further elaborated as a measure of molecular disarray or unpredictability within a system, as the productive energy originates from organized molecular motion [33,34]. The present study adopts the notion of entropy as presented in Shazia Manzoor’s research [35]. Distance-based entropy is widely utilized in industrial chemistry to calculate the complexity of molecules and molecular ensembles, electronic structures, signal processing, physicochemical processes, and so on [36,37]. Extensive research has been conducted on graph entropy using topological indices across diverse networks [36,38,39,40,41,42,43,44], where Arockiaraj et al. [38] examined triphenylene-based metal and covalent organic frameworks, and Raza et al. [39] investigated coronene-based transition metal organic frameworks. In this study, we compute $RR$, $R{M}_2$, J, $Z_{a,b}$, and $ESO$ indices, along with entropies, for three types of coronene fractal structures, namely ZHCF, AHCF, and RCF.

2. Degree-Based Entropy

Manzoor et al. [45] and Ghani et al. [36] recently proposed a novel strategy by incorporating Shannon entropy [30] in the context of topological indices. The degree-based entropy is defined as follows:

$$EN{T}_{\mu }(G)=-\sum _{uv\in E\left(G\right)}{\displaystyle \frac{\mu \left(uv\right)}{{\sum }_{uv\in E\left(G\right)}\mu (uv)}}log\left({\displaystyle \frac{\mu \left(uv\right)}{{\sum }_{uv\in E\left(G\right)}\mu (uv)}}\right).$$

(1)

where u and v are vertices of G, $E\left(G\right)$ is the edge set, and $\mu \left(uv\right)$ denotes the edge weight of edge $uv$. Below, we present entropy formulations corresponding to various topological indices.

• Entropy related to the reciprocal Randić index

For $\mu (uv)=\sqrt{d_u{d}_v}$, the entropy associated with the reciprocal Randić index is given by

$$EN{T}_{RR}(G)=log(RR(G))-{\displaystyle \frac{1}{RR\left(G\right)}}log\left(\prod _{uv\in E\left(G\right)}{\left(\sqrt{d_u{d}_v}\right)}^{\sqrt{d_u{d}_v}}\right).$$

(2)

• Entropy related to the reduced second Zagreb index

For $\mu (uv)=(d_u-1)(d_v-1)$, the entropy corresponding to the reduced second Zagreb index is

$$EN{T}_{R{M}_2}(G)=log(R{M}_2(G))-{\displaystyle \frac{1}{R{M}_2(G)}}log\left(\prod _{uv\in E\left(G\right)}{\left((d_u-1)(d_v-1)\right)}^{(d_u-1)(d_v-1)}\right).$$

(3)

• Entropy related to the Balaban index

For $\mu (uv)={\displaystyle \frac{m}{m-n+2}}{\displaystyle \frac{1}{\sqrt{d_u{d}_v}}}$, the entropy associated with the Balaban index is given by

$$EN{T}_J(G)=log(J(G))-{\displaystyle \frac{1}{J\left(G\right)}}log\left(\prod _{uv\in E\left(G\right)}{\left(\left({\displaystyle \frac{m}{m-n+2}}\right){\displaystyle \frac{1}{\sqrt{d_u{d}_v}}}\right)}^{\left({\displaystyle \frac{m}{m-n+2}}\right){\displaystyle \frac{1}{\sqrt{d_u{d}_v}}}}\right).$$

(4)

• Entropy related to the $(a,b)$-Zagreb index

For $\mu (uv)=d_u^a{d}_v^b+d_u^b{d}_v^a$, the entropy associated with the $(a,b)$-Zagreb index is

$$EN{T}_{Z_{a,b}}(G)=log(Z_{a,b}\left(G\right))-{\displaystyle \frac{1}{Z_{a,b}(G)}}log\left(\prod _{uv\in E\left(G\right)}{\left(d_u^a{d}_v^b+d_u^b{d}_v^a\right)}^{d_u^a{d}_v^b+d_u^b{d}_v^a}\right).$$

(5)

• Entropy related to the elliptic sombor index

For $\mu (uv)=(d_u+d_v)\sqrt{d_u^2+d_v^2}$, the entropy corresponding to the elliptic sombor index is

$$EN{T}_{ESO}(G)=log(ESO(G))-{\displaystyle \frac{1}{ESO\left(G\right)}}log\left(\prod _{uv\in E\left(G\right)}{\left((d_u+d_v)\sqrt{d_u^2+d_v^2}\right)}^{(d_u+d_v)\sqrt{d_u^2+d_v^2}}\right).$$

(6)

3. Topological Indices and Entropies of ZHCF, AHCF, and RCF

The coronene fractal structure is a planar circulene graph. Here, the vertices represent non-hydrogen atoms, and the edges represent covalent bonds between the corresponding atoms. The hydrogen atoms are not considered in our calculations. All edges are assumed to have a unit length of 1.

The t-dimensional zigzag and armchair hexagonal coronene fractals are denoted by $ZHCF\left(t\right)$ and $AHCF\left(t\right)$, respectively, as shown in Figure 2 and Figure 3. The rectangular coronene fractal with length p and breadth q is generated by arranging the fractal structures at a rectangular $p\times q$ phase, as in Figure 4, and denoted by $RCF(p,q)$. Notably, we have $ZHCF\left(1\right)=AHCF\left(1\right)=RCF(1,1)$. The coronene fractals $ZHCF\left(t\right)$, $AHCF\left(t\right)$, and $RCF(p,q)$ are constructed by repeating a fixed unit of coronene on different stages systematically. More specifically, $ZHCF\left(1\right)$, $ZHCF\left(2\right)$, and $ZHCF\left(3\right)$ can be constructed based on the three graphs in Figure 5, respectively. In particular, each vertex in the three graphs in Figure 5 corresponds to a coronene unit. Two coronenes share an edge if and only if the corresponding vertices in the original graphs are adjacent. Similarly, $AHCF\left(1\right)$, $AHCF\left(2\right)$, and $AHCF\left(3\right)$ correspond to the graphs in Figure 6, and $RCF(1,1)$, $RCF(4,3)$, and $RCF(8,3)$ correspond to those in Figure 7.

Moreover, for the construction of coronene fractal structures, we refer to [42].

Recall that $d_i$ denotes the degree of vertex i. Figure 8 illustrates the vertex degrees in the coronene fractal $ZHCF\left(1\right)$. In this section, we compute the indices and entropies of $RR$, $R{M}_2$, J, $Z_{a,b}$, and $ESO$ for the zigzag hexagonal coronene fractal, armchair hexagonal coronene fractal, and rectangular coronene fractal.

Theorem 1.

Let N be the zigzag hexagonal coronene fractal structure $ZHCF\left(t\right)$. Then,

1. 

$RR(N)=(387+36\sqrt{6})t^2+(12\sqrt{6}-33)t$;

2. 

$R{M}_2(N)=558t^2-30t$;

3. 

$J(N)={\displaystyle \frac{(8208+1026\sqrt{6})t^4+(360\sqrt{6}-198)t^3+(6\sqrt{6}-6)t^2}{45t^2-3t+2}}$;

4. 

$Z_{a,b}(N)=(18·2^{a+b+1}+36·2^a{3}^b+36·2^b{3}^a+234·3^{a+b})t^2+(6·2^{a+b+1}+12·2^a{3}^b+12·2^b{3}^a-30·3^{a+b})t$;

5. 

$ESO(N)=(2250\sqrt{2}+180\sqrt{13})t^2+(60\sqrt{13}-222\sqrt{2})t$.

Proof. 

By examining the graph structures depicted in Figure 2 and Figure 5, and through straightforward calculations, we can deduce that the numbers of vertices and edges of N are $126t^2+6t$ and $171t^2+3t$, respectively. Regarding the edge set of N, the possible degree pairs of the endpoints of each edge are exclusively $(2,2)$, $(2,3)$, or $(3,3)$. Consequently, the edge set of N can be partitioned into three subsets, with the number of edges in each subset detailed in

Table 1. Edge distribution of N based on the degree.

$({\mathit{d}}_{\mathit{u}},{\mathit{d}}_{\mathit{v}})$	Number of Edges
$(2,2)$	$18t^2+6t$
$(2,3)$	$36t^2+12t$
$(3,3)$	$117t^2-15t$

.

Using the values from Table 1 and the definitions of five topological indices, we obtain

$$\begin{array}{cc}\hfill RR\left(N\right)=& \sum _{uv\in E\left(N\right)}\sqrt{d_u{d}_v}\hfill \\ \hfill =& (18t^2+6t)\sqrt{2\times 2}+(36t^2+12t)\sqrt{2\times 3}+(117t^2-15t)\sqrt{3\times 3}\hfill \\ \hfill =& (387+36\sqrt{6})t^2+(12\sqrt{6}-33)t.\hfill \end{array}$$

$$\begin{array}{cc}\hfill R{M}_2(N)=& \sum _{uv\in E\left(N\right)}(d_u-1)(d_v-1)\hfill \\ \hfill =& (18t^2+6t)+(36t^2+12t)\times 2+(117t^2-15t)\times 4\hfill \\ \hfill =& 558t^2-30t.\hfill \end{array}$$

$$\begin{array}{cc}\hfill J\left(N\right)=& \sum _{uv\in E\left(N\right)}\left({\displaystyle \frac{m}{m-n+2}}\right){\displaystyle \frac{1}{\sqrt{d_u{d}_v}}}\hfill \\ \hfill =& {\displaystyle \frac{171t^2+3t}{45t^2-3t+2}}\left((18t^2+6t){\displaystyle \frac{1}{\sqrt{2\times 2}}}+(36t^2+12t){\displaystyle \frac{1}{\sqrt{2\times 3}}}+(117t^2-15t){\displaystyle \frac{1}{\sqrt{3\times 3}}}\right)\hfill \\ \hfill =& {\displaystyle \frac{(8208+1026\sqrt{6})t^4+(360\sqrt{6}-198)t^3+(6\sqrt{6}-6)t^2}{45t^2-3t+2}}.\hfill \end{array}$$

$$\begin{array}{cc}\hfill Z_{a,b}(N)=& \sum _{uv\in E\left(N\right)}\left(d_u^a{d}_v^b+d_u^b{d}_v^a\right)\hfill \\ \hfill =& (18t^2+6t)(2^a{2}^b+2^b{2}^a)+(36t^2+12t)(2^a{3}^b+2^b{3}^a)\hfill \\ \hfill & +(117t^2-15t)(3^a{3}^b+3^b{3}^a)\hfill \\ \hfill =& (18·2^{a+b+1}+36·2^a{3}^b+36·2^b{3}^a+234·3^{a+b})t^2\hfill \\ \hfill & +(6·2^{a+b+1}+12·2^a{3}^b+12·2^b{3}^a-30·3^{a+b})t.\hfill \end{array}$$

$$\begin{array}{cc}\hfill ESO\left(N\right)=& \sum _{uv\in E\left(N\right)}(d_u+d_v)\sqrt{d_u^2+d_v^2}\hfill \\ \hfill =& (18t^2+6t)(2+2)\sqrt{2^2+2^2}+(36t^2+12t)(2+3)\sqrt{2^2+3^2}\hfill \\ \hfill & +(117t^2-15t)(3+3)\sqrt{3^2+3^2}\hfill \\ \hfill =& (2250\sqrt{2}+180\sqrt{13})t^2+(60\sqrt{13}-222\sqrt{2})t.\hfill \end{array}$$

□

Theorem 2.

Let N be the zigzag hexagonal coronene fractal structure $ZHCF\left(t\right)$. Then,

1. 

$EN{T}_{RR}(N)=log(RR(N))-{\displaystyle \frac{1}{RR\left(N\right)}}log\left(2^{36t^2+12t}\times {\sqrt{6}}^{36\sqrt{6}{t}^2+12\sqrt{6}t}\times 3^{351t^2-45t}\right)$;

2. 

$EN{T}_{R{M}_2}(N)=log(R{M}_2(N))-{\displaystyle \frac{1}{R{M}_2(N)}}log\left(2^{1008t^2-96t}\right)$;

3. 

$\begin{array}{cc}EN{T}_J(N)=\hfill & log(J(N))-{\displaystyle \frac{1}{J\left(N\right)}}log({\left({\displaystyle \frac{171t^2+3t}{2\times (45t^2-3t+2)}}\right)}^{{\displaystyle \frac{(171t^2+3t)\times (9t^2+3t)}{(45t^2-3t+2)}}}\hfill \\ & \times {\left({\displaystyle \frac{171t^2+3t}{\sqrt{6}\times (45t^2-3t+2)}}\right)}^{{\displaystyle \frac{(171t^2+3t)\times (6\sqrt{6}{t}^2+2\sqrt{6}t)}{(45t^2-3t+2)}}}\hfill \\ & \times {\left({\displaystyle \frac{171t^2+3t}{3\times (45t^2-3t+2)}}\right)}^{{\displaystyle \frac{(171t^2+3t)\times (39t^2-5t)}{(45t^2-3t+2)}}});\hfill \end{array}$

4. 

$\begin{array}{cc}EN{T}_{Z_{a,b}}(N)=\hfill & log(Z_{a,b}(N))-{\displaystyle \frac{1}{Z_{a,b}(N)}}log({\left(2^{a+b+1}\right)}^{2^{a+b+1}\times (18t^2+6t)}\hfill \\ & \times {\left(2^a{3}^b+2^b{3}^a\right)}^{(2^a{3}^b+2^b{3}^a)\times (36t^2+12t)}\hfill \\ & \times {\left(2\times 3^{a+b}\right)}^{(2\times 3^{a+b})\times (117t^2-15t)});\hfill \end{array}$

5. 

$\begin{array}{cc}EN{T}_{ESO}(N)=\hfill & log(ESO(N))-{\displaystyle \frac{1}{ESO\left(N\right)}}log({\left(8\sqrt{2}\right)}^{8\sqrt{2}(18t^2+6t)}\hfill \\ & \times {\left(5\sqrt{13}\right)}^{5\sqrt{13}(36t^2+12t)}\hfill \\ & \times {\left(18\sqrt{2}\right)}^{18\sqrt{2}(117t^2-15t)}).\hfill \end{array}$

Proof. 

To prove this, we refer to Table 1, which provides the edge distribution of N. Then, by Equations (2)–(6) and Theorem 1, we have

$$\begin{array}{cc}\hfill EN{T}_{RR}(N)& =log(RR(N))-{\displaystyle \frac{1}{RR\left(N\right)}}log\left(2^{2\times (18t^2+6t)}\times {\sqrt{6}}^{\sqrt{6}(36t^2+12t)}\times 3^{3\times (117t^2-15t)}\right)\hfill \\ & =log(RR(N))-{\displaystyle \frac{1}{RR\left(N\right)}}log\left(2^{36t^2+12t}\times {\sqrt{6}}^{36\sqrt{6}{t}^2+12\sqrt{6}t}\times 3^{351t^2-45t}\right).\hfill \\ \hfill EN{T}_{R{M}_2}(N)=& log(R{M}_2(N))-{\displaystyle \frac{1}{R{M}_2\left(N\right)}}log\left(1^{1\times (18t^2+6t)}\times 2^{2(36t^2+12t)}\times 4^{4(117t^2-15t)}\right)\hfill \\ & =log(R{M}_2(N))-{\displaystyle \frac{1}{R{M}_2\left(N\right)}}log\left(2^{1008t^2-96t}\right).\hfill \\ \hfill EN{T}_J(N)=& log(J(N))-{\displaystyle \frac{1}{J\left(N\right)}}log({\left({\displaystyle \frac{171t^2+3t}{2\times (45t^2-3t+2)}}\right)}^{{\displaystyle \frac{(171t^2+3t)\times (18t^2+6t)}{2(45t^2-3t+2)}}}\hfill \\ & \times {\left({\displaystyle \frac{171t^2+3t}{\sqrt{6}\times (45t^2-3t+2)}}\right)}^{{\displaystyle \frac{(171t^2+3t)\times (36t^2+12t)}{\sqrt{6}\times (45t^2-3t+2)}}}\times {\left({\displaystyle \frac{171t^2+3t}{3\times (45t^2-3t+2)}}\right)}^{{\displaystyle \frac{(171t^2+3t)\times (117t^2-15t)}{3\times (45t^2-3t+2)}}})\hfill \\ \hfill =& log\left(J\left(N\right)\right)-{\displaystyle \frac{1}{J\left(N\right)}}log({\left({\displaystyle \frac{171t^2+3t}{2\times (45t^2-3t+2)}}\right)}^{{\displaystyle \frac{(171t^2+3t)\times (9t^2+3t)}{(45t^2-3t+2)}}}\hfill \\ & \times {\left({\displaystyle \frac{171t^2+3t}{\sqrt{6}\times (45t^2-3t+2)}}\right)}^{{\displaystyle \frac{(171t^2+3t)\times (6\sqrt{6}{t}^2+2\sqrt{6}t)}{(45t^2-3t+2)}}}\times {\left({\displaystyle \frac{171t^2+3t}{3\times (45t^2-3t+2)}}\right)}^{{\displaystyle \frac{(171t^2+3t)\times (39t^2-5t)}{(45t^2-3t+2)}}}).\hfill \\ \hfill EN{T}_{Z_{a,b}}\left(N\right)=& log\left(Z_{a,b}\left(N\right)\right)-{\displaystyle \frac{1}{Z_{a,b}\left(N\right)}}log({\left(2^a{2}^b+2^b{2}^a\right)}^{(2^a{2}^b+2^b{2}^a)\times (18t^2+6t)}\hfill \\ & \times {\left(2^a{3}^b+2^b{3}^a\right)}^{(2^a{3}^b+2^b{3}^a)\times (36t^2+12t)}\hfill \\ & \times {\left(3^a{3}^b+3^b{3}^a\right)}^{(3^a{3}^b+3^b{3}^a)\times (117t^2-15t)})\hfill \\ \hfill =& log(Z_{a,b}(N))-{\displaystyle \frac{1}{Z_{a,b}\left(N\right)}}log({\left(2^{a+b+1}\right)}^{2^{a+b+1}\times (18t^2+6t)}\hfill \\ & \times {\left(2^a{3}^b+2^b{3}^a\right)}^{(2^a{3}^b+2^b{3}^a)\times (36t^2+12t)}\times {\left(2\times 3^{a+b}\right)}^{(2\times 3^{a+b})\times (117t^2-15t)}).\hfill \\ \hfill EN{T}_{ESO}\left(N\right)=& log(ESO(N))-{\displaystyle \frac{1}{ESO\left(N\right)}}log({\left((2+2)\sqrt{2^2+2^2}\right)}^{\left((2+2)\sqrt{2^2+2^2}\right)(18t^2+6t)}\hfill \\ & \times {\left((2+3)\sqrt{2^2+3^2}\right)}^{\left((2+3)\sqrt{2^2+3^2}\right)(36t^2+12t)}\hfill \\ & \times {\left((3+3)\sqrt{3^2+3^2}\right)}^{\left((3+3)\sqrt{3^2+3^2}\right)(117t^2-15t)})\hfill \\ \hfill =& log\left(ESO\left(N\right)\right)-{\displaystyle \frac{1}{ESO\left(N\right)}}log({\left(8\sqrt{2}\right)}^{8\sqrt{2}(18t^2+6t)}\times {\left(5\sqrt{13}\right)}^{5\sqrt{13}(36t^2+12t)}\hfill \\ & \times {\left(18\sqrt{2}\right)}^{18\sqrt{2}(117t^2-15t)}).\hfill \end{array}$$

□

Theorem 3.

Let P be the armchair coronene fractal structure $AHCF\left(t\right)$. Then,

1. 

$RR(P)=(1161+108\sqrt{6})t^2-(1227+84\sqrt{6})t+420+24\sqrt{6}$;

2. 

$R{M}_2(P)=1674t^2-1734t+588$;

3. 

$J(P)=((73872+9234\sqrt{6})t^4-(16308\sqrt{6}+148932)t^3+(12174\sqrt{6}+124878)t^2-(4380\sqrt{6}+50214)t+672\sqrt{6}+8400)/(135t^2-141t+50)$;

4. 

$Z_{a,b}(P)=(54·2^{a+b+1}+108·2^a{3}^b+108·2^b{3}^a+702·3^{a+b})t^2-(42·2^{a+b+1}+84·2^a{3}^b+84·2^b{3}^a+762·3^{a+b})t+12·2^{a+b+1}+24·2^a{3}^b+24·2^b{3}^a+264·3^{a+b}$;

5. 

$ESO(P)=(6750\sqrt{2}+540\sqrt{13})t^2-(7194\sqrt{2}+420\sqrt{13})t+2472\sqrt{2}+120\sqrt{13}$.

Proof. 

By examining the graph structures depicted in Figure 3 and Figure 6, and through straightforward calculations, we can deduce that the numbers of vertices and edges of P are $378t^2-366t+120$ and $513t^2-507t+168$, respectively. Regarding the edge set of N, the possible degree pairs of the endpoints of each edge are exclusively $(2,2)$, $(2,3)$, or $(3,3)$. Consequently, the edge set of N can be partitioned into three subsets, with the number of edges in each subset detailed in

Table 2. Edge distribution of P based on the degree.

$({\mathit{d}}_{\mathit{u}},{\mathit{d}}_{\mathit{v}})$	Number of Edges
$(2,2)$	$54t^2-42t+12$
$(2,3)$	$108t^2-84t+24$
$(3,3)$	$351t^2-381t+132$

.

Using the values from Table 2 and the definitions of five topological indices, we obtain

$$\begin{array}{cc}\hfill RR\left(P\right)=& \sum _{uv\in E\left(P\right)}\sqrt{d_u{d}_v}\hfill \\ \hfill =& (54t^2-42t+12)\sqrt{2\times 2}+(108t^2-84t+24)\sqrt{2\times 3}\hfill \\ \hfill & +(351t^2-381t+132)\sqrt{3\times 3}\hfill \\ \hfill =& (1161+108\sqrt{6})t^2-(1227+84\sqrt{6})t+420+24\sqrt{6}.\hfill \end{array}$$

$$\begin{array}{cc}\hfill R{M}_2(P)=& \sum _{uv\in E\left(P\right)}(d_u-1)(d_v-1)\hfill \\ \hfill =& (54t^2-42t+12)+(108t^2-84t+24)\times 2+(351t^2-381t+132)\times 4\hfill \\ \hfill =& 1674t^2-1734t+588.\hfill \end{array}$$

$$\begin{array}{cc}\hfill J\left(P\right)=& \sum _{uv\in E\left(P\right)}\left({\displaystyle \frac{m}{m-n+2}}\right){\displaystyle \frac{1}{\sqrt{d_u{d}_v}}}\hfill \\ \hfill =& {\displaystyle \frac{513t^2-507t+168}{135t^2-141t+50}}((54t^2-42t+12){\displaystyle \frac{1}{2}}+(108t^2-84t+24){\displaystyle \frac{1}{\sqrt{6}}}\hfill \\ \hfill & +(351t^2-381t+132){\displaystyle \frac{1}{3}})\hfill \\ \hfill =& ((73872+9234\sqrt{6})t^4-(16308\sqrt{6}+148932)t^3+(12174\sqrt{6}+124878)t^2\hfill \\ \hfill & -(4380\sqrt{6}+50214)t+672\sqrt{6}+8400)/(135t^2-141t+50).\hfill \end{array}$$

$$\begin{array}{cc}\hfill Z_{a,b}(P)=& \sum _{uv\in E\left(P\right)}\left(d_u^a{d}_v^b+d_u^b{d}_v^a\right)\hfill \\ \hfill =& (54t^2-42t+12)\left(2^{a+b+1}\right)+(108t^2-84t+24)(2^a{3}^b+2^b{3}^a)\hfill \\ \hfill & +(351t^2-381t+132)(2\times 3^{a+b})\hfill \\ \hfill =& (54·2^{a+b+1}+108·2^a{3}^b+108·2^b{3}^a+702·3^{a+b})t^2\hfill \\ \hfill & -(42·2^{a+b+1}+84·2^a{3}^b+84·2^b{3}^a+762·3^{a+b})t\hfill \\ \hfill & +12·2^{a+b+1}+24·2^a{3}^b+24·2^b{3}^a+264·3^{a+b}.\hfill \end{array}$$

$$\begin{array}{cc}\hfill ESO\left(P\right)=& \sum _{uv\in E\left(P\right)}(d_u+d_v)\sqrt{d_u^2+d_v^2}\hfill \\ \hfill =& (54t^2-42t+12)(2+2)\sqrt{2^2+2^2}+(108t^2-84t+24)(2+3)\sqrt{2^2+3^2}\hfill \\ \hfill & +(351t^2-381t+132)(3+3)\sqrt{3^2+3^2}\hfill \\ \hfill =& (6750\sqrt{2}+540\sqrt{13})t^2-(7194\sqrt{2}+420\sqrt{13})t+2472\sqrt{2}+120\sqrt{13}.\hfill \end{array}$$

□

Theorem 4.

Let P be the armchair coronene fractal structure $AHCF\left(t\right)$. Then,

1. 

$\begin{array}{cc}EN{T}_{RR}(P)=\hfill & log(RR(P))-{\displaystyle \frac{1}{RR\left(P\right)}}log(2^{108t^2-84t+24}\times {\sqrt{6}}^{108\sqrt{6}{t}^2-84\sqrt{6}t+24\sqrt{6}}\hfill \\ & \times 3^{1053t^2-1143t+396});\hfill \end{array}$

2. 

$EN{T}_{R{M}_2}(P)=log(R{M}_2(P))-{\displaystyle \frac{1}{R{M}_2\left(P\right)}}log\left(2^{3024t^2-3216t+1104}\right)$;

3. 

$\begin{array}{c}EN{T}_J(P)=log(J(P))-{\displaystyle \frac{1}{J\left(P\right)}}log({\left({\displaystyle \frac{513t^2-507t+168}{2\times (135t^2-141t+50)}}\right)}^{{\displaystyle \frac{(513t^2-507t+168)\times (27t^2-21t+6)}{(135t^2-141t+50)}}}\hfill \\ & \times {\left({\displaystyle \frac{513t^2-507t+168}{\sqrt{6}\times (135t^2-141t+50)}}\right)}^{{\displaystyle \frac{(513t^2-507t+168)\times (18\sqrt{6}{t}^2-14\sqrt{6}t+4\sqrt{6})}{(135t^2-141t+50)}}}\hfill \\ & \times {\left({\displaystyle \frac{513t^2-507t+168}{3\times (135t^2-141t+50)}}\right)}^{{\displaystyle \frac{(513t^2-507t+168)\times (117t^2-127t+44)}{(135t^2-141t+50)}}});\hfill \end{array}$

4. 

$\begin{array}{c}EN{T}_{Z_{a,b}}(P)=log(Z_{a,b}(P))-{\displaystyle \frac{1}{Z_{a,b}(P)}}log({\left(2^{a+b+1}\right)}^{2^{a+b+1}\times (54t^2-42t+12)}\hfill \\ & \times {\left(2^a{3}^b+2^b{3}^a\right)}^{(2^a{3}^b+2^b{3}^a)\times (108t^2-84t+24)}\hfill \\ & \times {\left(2\times 3^{a+b}\right)}^{(2\times 3^{a+b})\times (351t^2-381t+132)});\hfill \end{array}$

5. 

$\begin{array}{c}EN{T}_{ESO}(P)=log(ESO(P))-{\displaystyle \frac{1}{ESO\left(P\right)}}log({\left(8\sqrt{2}\right)}^{8\sqrt{2}(54t^2-42t+12)}\hfill \\ & \times {\left(5\sqrt{13}\right)}^{5\sqrt{13}(108t^2-84t+24)}\hfill \\ & \times {\left(18\sqrt{2}\right)}^{18\sqrt{2}(351t^2-381t+132)}).\hfill \end{array}$

Proof. 

To prove this, we refer to Table 2, which provides the edge distribution of P. Then, using Equations (2)–(6) and Theorem 3, we have

$$\begin{array}{cc}\hfill EN{T}_{RR}(P)=& log(RR(P))-{\displaystyle \frac{1}{RR\left(P\right)}}log(2^{2\times (54t^2-42t+12)}\times {\sqrt{6}}^{\sqrt{6}(108t^2-84t+24)}\hfill \\ \hfill & \times 3^{3\times (351t^2-381t+132)})\hfill \\ \hfill =& log(RR(P))-{\displaystyle \frac{1}{RR\left(P\right)}}log(2^{108t^2-84t+24}\times {\sqrt{6}}^{108\sqrt{6}{t}^2-84\sqrt{6}t+24\sqrt{6}}\hfill \\ \hfill & \times 3^{1053t^2-1143t+396}).\hfill \end{array}$$

$$\begin{array}{cc}\hfill EN{T}_{R{M}_2}(P)=& log(R{M}_2(P))-{\displaystyle \frac{1}{R{M}_2(P)}}log(1^{1\times (54t^2-42t+12)}\times 2^{2(108t^2-84t+24)}\hfill \\ \hfill & \times 4^{4(351t^2-381t+132)})\hfill \\ \hfill =& log(R{M}_2(P))-{\displaystyle \frac{1}{R{M}_2(P)}}log\left(2^{3024t^2-3216t+1104}\right).\hfill \end{array}$$

$$\begin{array}{cc}\hfill EN{T}_J(P)=& log(J(P))-{\displaystyle \frac{1}{J\left(P\right)}}log({\left({\displaystyle \frac{513t^2-507t+168}{2\times (135t^2-141t+50)}}\right)}^{{\displaystyle \frac{(513t^2-507t+168)\times (54t^2-42t+12)}{2\times (135t^2-141t+50)}}}\hfill \\ & \times {\left({\displaystyle \frac{513t^2-507t+168}{\sqrt{6}\times (135t^2-141t+50)}}\right)}^{{\displaystyle \frac{(513t^2-507t+168)\times (108t^2-84t+24)}{\sqrt{6}\times (135t^2-141t+50)}}}\hfill \\ & \times {\left({\displaystyle \frac{513t^2-507t+168}{3\times (135t^2-141t+50)}}\right)}^{{\displaystyle \frac{(513t^2-507t+168)\times (351t^2-381t+132)}{3\times (135t^2-141t+50)}}})\hfill \\ \hfill =& log(J(P))-{\displaystyle \frac{1}{J\left(P\right)}}log({\left({\displaystyle \frac{513t^2-507t+168}{2\times (135t^2-141t+50)}}\right)}^{{\displaystyle \frac{(513t^2-507t+168)\times (27t^2-21t+6)}{(135t^2-141t+50)}}}\hfill \\ & \times {\left({\displaystyle \frac{513t^2-507t+168}{\sqrt{6}\times (135t^2-141t+50)}}\right)}^{{\displaystyle \frac{(513t^2-507t+168)\times (18\sqrt{6}{t}^2-14\sqrt{6}t+4\sqrt{6})}{(135t^2-141t+50)}}}\hfill \\ & \times {\left({\displaystyle \frac{513t^2-507t+168}{3\times (135t^2-141t+50)}}\right)}^{{\displaystyle \frac{(513t^2-507t+168)\times (117t^2-127t+44)}{(135t^2-141t+50)}}}).\hfill \end{array}$$

$$\begin{array}{cc}\hfill EN{T}_{Z_{a,b}}(P)=& log(Z_{a,b}(P))-{\displaystyle \frac{1}{Z_{a,b}(P)}}log({\left(2^a{2}^b+2^b{2}^a\right)}^{(2^a{2}^b+2^b{2}^a)\times (54t^2-42t+12)}\hfill \\ & \times {\left(2^a{3}^b+2^b{3}^a\right)}^{(2^a{3}^b+2^b{3}^a)\times (108t^2-84t+24)}\hfill \\ & \times {\left(3^a{3}^b+3^b{3}^a\right)}^{(3^a{3}^b+3^b{3}^a)\times (351t^2-381t+132)})\hfill \\ \hfill =& log(Z_{a,b}(P))-{\displaystyle \frac{1}{Z_{a,b}(P)}}log({\left(2^{a+b+1}\right)}^{2^{a+b+1}\times (54t^2-42t+12)}\hfill \\ & \times {\left(2^a{3}^b+2^b{3}^a\right)}^{(2^a{3}^b+2^b{3}^a)\times (108t^2-84t+24)}\hfill \\ & \times {\left(2\times 3^{a+b}\right)}^{(2\times 3^{a+b})\times (351t^2-381t+132)}).\hfill \end{array}$$

$$\begin{array}{cc}\hfill EN{T}_{ESO}(P)=& log(ESO(P))-{\displaystyle \frac{1}{ESO\left(P\right)}}log({\left((2+2)\sqrt{2^2+2^2}\right)}^{\left((2+2)\sqrt{2^2+2^2}\right)(54t^2-42t+12)}\hfill \\ & \times {\left((2+3)\sqrt{2^2+3^2}\right)}^{\left((2+3)\sqrt{2^2+3^2}\right)(108t^2-84t+24)}\hfill \\ & \times {\left((3+3)\sqrt{3^2+3^2}\right)}^{\left((3+3)\sqrt{3^2+3^2}\right)(351t^2-381t+132)})\hfill \\ \hfill =& log(ESO(P))-{\displaystyle \frac{1}{ESO\left(P\right)}}log({\left(8\sqrt{2}\right)}^{8\sqrt{2}(54t^2-42t+12)}\hfill \\ & \times {\left(5\sqrt{13}\right)}^{5\sqrt{13}(108t^2-84t+24)}\hfill \\ & \times {\left(18\sqrt{2}\right)}^{18\sqrt{2}(351t^2-381t+132)}).\hfill \end{array}$$

□

Theorem 5.

Let R be the rectangular coronene fractal structure $RCF(p,q)$. Then,

1. 

$RR(R)=(258+24\sqrt{6})pq+(4\sqrt{6}-11)p+(107+20\sqrt{6})q$;

2. 

$R{M}_2(R)=372pq-10p+166q$;

3. 

$J(R)={\displaystyle \frac{(114pq+p+59q)((32+4\sqrt{6})pq+\frac{2\sqrt{6}-2}{3}p+\frac{10\sqrt{6}+44}{3}q)}{30pq-p+13q+2}}$;

4. 

$Z_{a,b}(R)=(12·2^{a+b+1}+24·2^a{3}^b+24·2^b{3}^a+156·3^{a+b})pq+(2^{a+b+2}+2^{a+2}{3}^b+2^{b+2}{3}^a-10·3^{a+b})p+(10·2^{a+b+1}+20·2^a{3}^b+20·2^b{3}^a+58·3^{a+b})q$;

5. 

$ESO(R)=(1500\sqrt{2}+120\sqrt{13})pq+(20\sqrt{13}-74\sqrt{2})p+(602\sqrt{2}+100\sqrt{13})q$.

Proof. 

By examining the graph structures depicted in Figure 4 and Figure 7, and through straightforward calculations, we can deduce that the numbers of vertices and edges of R are $84pq+2p+46q$ and $114pq+p+59q$, respectively. Regarding the edge set of N, the possible degree pairs of the endpoints of each edge are exclusively $(2,2)$, $(2,3)$, or $(3,3)$. Consequently, the edge set of N can be partitioned into three subsets, with the number of edges in each subset detailed in

Table 3. Edge distribution of R based on the degree.

$({\mathit{d}}_{\mathit{u}},{\mathit{d}}_{\mathit{v}})$	Number of Edges
$(2,2)$	$12pq+2p+10q$
$(2,3)$	$24pq+4p+20q$
$(3,3)$	$78pq-5p+29q$

.

Using the values from Table 3 and the definitions of five topological indices, we obtain

$$\begin{array}{cc}\hfill RR\left(R\right)=& \sum _{uv\in E\left(R\right)}\sqrt{d_u{d}_v}\hfill \\ \hfill =& (12pq+2p+10q)\sqrt{2\times 2}+(24pq+4p+20q)\sqrt{2\times 3}\hfill \\ \hfill & +(78pq-5p+29q)\sqrt{3\times 3}\hfill \\ \hfill =& (258+24\sqrt{6})pq+(4\sqrt{6}-11)p+(107+20\sqrt{6})q.\hfill \end{array}$$

$$\begin{array}{cc}\hfill R{M}_2(R)=& \sum _{uv\in E\left(R\right)}(d_u-1)(d_v-1)\hfill \\ \hfill =& (12pq+2p+10q)+(24pq+4p+20q)\times 2+(78pq-5p+29q)\times 4\hfill \\ \hfill =& 372pq-10p+166q.\hfill \end{array}$$

$$\begin{array}{cc}\hfill J\left(R\right)=& \sum _{uv\in E\left(R\right)}\left({\displaystyle \frac{m}{m-n+2}}\right){\displaystyle \frac{1}{\sqrt{d_u{d}_v}}}\hfill \\ \hfill =& {\displaystyle \frac{114pq+p+59q}{30pq-p+13q+2}}((12pq+2p+10q){\displaystyle \frac{1}{2}}+(24pq+4p+20q){\displaystyle \frac{1}{\sqrt{6}}}\hfill \\ \hfill & +(78pq-5p+29q){\displaystyle \frac{1}{3}})\hfill \\ \hfill =& {\displaystyle \frac{(114pq+p+59q)((32+4\sqrt{6})pq+\frac{2\sqrt{6}-2}{3}p+\frac{10\sqrt{6}+44}{3}q)}{30pq-p+13q+2}}.\hfill \end{array}$$

$$\begin{array}{cc}\hfill Z_{a,b}(R)=& \sum _{uv\in E\left(R\right)}\left(d_u^a{d}_v^b+d_u^b{d}_v^a\right)\hfill \\ \hfill =& (12pq+2p+10q)\left(2^{a+b+1}\right)+(24pq+4p+20q)(2^a{3}^b+2^b{3}^a)\hfill \\ \hfill & +(78pq-5p+29q)(2\times 3^{a+b})\hfill \\ \hfill =& (12·2^{a+b+1}+24·2^a{3}^b+24·2^b{3}^a+156·3^{a+b})pq\hfill \\ \hfill & +(2^{a+b+2}+2^{a+2}{3}^b+2^{b+2}{3}^a-10·3^{a+b})p\hfill \\ \hfill & +(10·2^{a+b+1}+20·2^a{3}^b+20·2^b{3}^a+58·3^{a+b})q.\hfill \end{array}$$

$$\begin{array}{cc}\hfill ESO\left(R\right)=& \sum _{uv\in E\left(R\right)}(d_u+d_v)\sqrt{d_u^2+d_v^2}\hfill \\ \hfill =& (12pq+2p+10q)(2+2)\sqrt{2^2+2^2}+(24pq+4p+20q)(2+3)\sqrt{2^2+3^2}\hfill \\ \hfill & +(78pq-5p+29q)(3+3)\sqrt{3^2+3^2}\hfill \\ \hfill =& (1500\sqrt{2}+120\sqrt{13})pq+(20\sqrt{13}-74\sqrt{2})p+(602\sqrt{2}+100\sqrt{13})q.\hfill \end{array}$$

□

Theorem 6.

Let R be the rectangular coronene fractal structure $RCF(p,q)$. Then,

1. 

$\begin{array}{cc}EN{T}_{RR}(R)=\hfill & log(RR(R))-{\displaystyle \frac{1}{RR\left(R\right)}}log(2^{24pq+4p+20q}\times {\sqrt{6}}^{24\sqrt{6}pq+4\sqrt{6}p+20\sqrt{6}q}\hfill \\ & \times 3^{234pq-15p+87q});\hfill \end{array}$

2. 

$EN{T}_{R{M}_2}(R)=log(R{M}_2(R))-{\displaystyle \frac{1}{R{M}_2\left(R\right)}}log\left(2^{672pq-32p+272q}\right)$;

3. 

$\begin{array}{cc}EN{T}_J\left(R\right)=\hfill & log\left(J\left(R\right)\right)-{\displaystyle \frac{1}{J\left(R\right)}}log({\left({\displaystyle \frac{114pq+p+59q}{2\times (30pq-p+13q+2)}}\right)}^{{\displaystyle \frac{(114pq+p+59q)\times (6pq+p+5q)}{(30pq-p+13q+2)}}}\hfill \\ & \times {\left({\displaystyle \frac{114pq+p+59q}{\sqrt{6}\times (30pq-p+13q+2)}}\right)}^{{\displaystyle \frac{(114pq+p+59q)\times (4\sqrt{6}pq+\frac{2\sqrt{6}}{3}p+\frac{10\sqrt{6}}{3}q)}{(30pq-p+13q+2)}}}\hfill \\ & \times {\left({\displaystyle \frac{114pq+p+59q}{3\times (30pq-p+13q+2)}}\right)}^{{\displaystyle \frac{(114pq+p+59q)\times (26pq-\frac{5}{3}p+\frac{29}{3}q)}{(30pq-p+13q+2)}}});\hfill \end{array}$

4. 

$\begin{array}{cc}EN{T}_{Z_{a,b}}(R)=\hfill & log(Z_{a,b}(R))-{\displaystyle \frac{1}{Z_{a,b}(R)}}log({\left(2^{a+b+1}\right)}^{2^{a+b+1}\times (12pq+2p+10q)}\hfill \\ & \times {\left(2^a{3}^b+2^b{3}^a\right)}^{(2^a{3}^b+2^b{3}^a)\times (24pq+4p+20q)}\hfill \\ & \times {\left(2\times 3^{a+b}\right)}^{(2\times 3^{a+b})\times (78pq-5p+29q)});\hfill \end{array}$

5. 

$\begin{array}{cc}EN{T}_{ESO}(R)=\hfill & log(ESO(R))-{\displaystyle \frac{1}{ESO\left(R\right)}}log({\left(8\sqrt{2}\right)}^{8\sqrt{2}(12pq+2p+10q)}\hfill \\ & \times {\left(5\sqrt{13}\right)}^{5\sqrt{13}(24pq+4p+20q)}\hfill \\ & \times {\left(18\sqrt{2}\right)}^{18\sqrt{2}(78pq-5p+29q)}).\hfill \end{array}$

Proof. 

To prove this, we refer to Table 3, which provides the edge distribution of R. Then, using Equations (2)–(6) and Theorem 5, we have

$$\begin{array}{cc}\hfill EN{T}_{RR}(R)=& log(RR(R))-{\displaystyle \frac{1}{RR\left(R\right)}}log(2^{2\times (12pq+2p+10q)}\times {\sqrt{6}}^{\sqrt{6}(24pq+4p+20q)}\hfill \\ \hfill & \times 3^{3\times (78pq-5p+29q)})\hfill \\ \hfill =& log(RR(R))-{\displaystyle \frac{1}{RR\left(R\right)}}log(2^{24pq+4p+20q}\times {\sqrt{6}}^{24\sqrt{6}pq+4\sqrt{6}p+20\sqrt{6}q}\hfill \\ \hfill & \times 3^{234pq-15p+87q}).\hfill \end{array}$$

$$\begin{array}{cc}\hfill EN{T}_{R{M}_2}(R)=& log(R{M}_2(R))-{\displaystyle \frac{1}{R{M}_2(R)}}log(1^{1\times (12pq+2p+10q)}\times 2^{2(24pq+4p+20q)}\hfill \\ \hfill & \times 4^{4(78pq-5p+29q)})\hfill \\ \hfill =& log(R{M}_2(R))-{\displaystyle \frac{1}{R{M}_2\left(R\right)}}log\left(2^{672pq-32p+272q}\right).\hfill \end{array}$$

$$\begin{array}{cc}\hfill EN{T}_J(R)=& log(J(R))-{\displaystyle \frac{1}{J\left(R\right)}}log({\left({\displaystyle \frac{114pq+p+59q}{2\times (30pq-p+13q+2)}}\right)}^{{\displaystyle \frac{(114pq+p+59q)\times (12pq+2p+10q)}{2(30pq-p+13q+2)}}}\hfill \\ & \times {\left({\displaystyle \frac{114pq+p+59q}{\sqrt{6}\times (30pq-p+13q+2)}}\right)}^{{\displaystyle \frac{(114pq+p+59q)\times (24pq+4p+20q)}{\sqrt{6}\times (30pq-p+13q+2)}}}\hfill \\ & \times {\left({\displaystyle \frac{114pq+p+59q}{3\times (30pq-p+13q+2)}}\right)}^{{\displaystyle \frac{(114pq+p+59q)\times (78pq-5p+29q)}{3\times (30pq-p+13q+2)}}})\hfill \\ \hfill =& log\left(J\left(R\right)\right)-{\displaystyle \frac{1}{J\left(R\right)}}log({\left({\displaystyle \frac{114pq+p+59q}{2\times (30pq-p+13q+2)}}\right)}^{{\displaystyle \frac{(114pq+p+59q)\times (6pq+p+5q)}{(30pq-p+13q+2)}}}\hfill \\ & \times {\left({\displaystyle \frac{114pq+p+59q}{\sqrt{6}\times (30pq-p+13q+2)}}\right)}^{{\displaystyle \frac{(114pq+p+59q)\times (4\sqrt{6}pq+\frac{2\sqrt{6}}{3}p+\frac{10\sqrt{6}}{3}q)}{(30pq-p+13q+2)}}}\hfill \\ & \times {\left({\displaystyle \frac{114pq+p+59q}{3\times (30pq-p+13q+2)}}\right)}^{{\displaystyle \frac{(114pq+p+59q)\times (26pq-\frac{5}{3}p+\frac{29}{3}q)}{(30pq-p+13q+2)}}}).\hfill \end{array}$$

$$\begin{array}{cc}\hfill EN{T}_{Z_{a,b}}(R)=& log(Z_{a,b}(R))-{\displaystyle \frac{1}{Z_{a,b}(R)}}log({\left(2^a{2}^b+2^b{2}^a\right)}^{(2^a{2}^b+2^b{2}^a)\times (12pq+2p+10q)}\hfill \\ & \times {\left(2^a{3}^b+2^b{3}^a\right)}^{(2^a{3}^b+2^b{3}^a)\times (24pq+4p+20q)}\hfill \\ & \times {\left(3^a{3}^b+3^b{3}^a\right)}^{(3^a{3}^b+3^b{3}^a)\times (78pq-5p+29q)})\hfill \\ \hfill =& log\left(Z_{a,b}\left(R\right)\right)-{\displaystyle \frac{1}{Z_{a,b}\left(R\right)}}log({\left(2^{a+b+1}\right)}^{2^{a+b+1}\times (12pq+2p+10q)}\hfill \\ & \times {\left(2^a{3}^b+2^b{3}^a\right)}^{(2^a{3}^b+2^b{3}^a)\times (24pq+4p+20q)}\hfill \\ & \times {\left(2\times 3^{a+b}\right)}^{(2\times 3^{a+b})\times (78pq-5p+29q)}).\hfill \end{array}$$

$$\begin{array}{cc}\hfill EN{T}_{ESO}(R)=& log(ESO)R))-{\displaystyle \frac{1}{ESO\left(R\right)}}log({\left((2+2)\sqrt{2^2+2^2}\right)}^{\left((2+2)\sqrt{2^2+2^2}\right)(12pq+2p+10q)}\hfill \\ & \times {\left((2+3)\sqrt{2^2+3^2}\right)}^{\left((2+3)\sqrt{2^2+3^2}\right)(24pq+4p+20q)}\hfill \\ & \times {\left((3+3)\sqrt{3^2+3^2}\right)}^{\left((3+3)\sqrt{3^2+3^2}\right)(78pq-5p+29q)})\hfill \\ \hfill =& log(ESO(R))-{\displaystyle \frac{1}{ESO\left(R\right)}}log({\left(8\sqrt{2}\right)}^{8\sqrt{2}(12pq+2p+10q)}\hfill \\ & \times {\left(5\sqrt{13}\right)}^{5\sqrt{13}(24pq+4p+20q)}\hfill \\ & \times {\left(18\sqrt{2}\right)}^{18\sqrt{2}(78pq-5p+29q)}).\hfill \end{array}$$

□

4. Comparison

In this section, we numerically and graphically compare the entropies related to $RR$, $R{M}_2$, J, $Z_{3,3}$, and $ESO$ for $ZHCF\left(t\right)$, $AHCF\left(t\right)$, and $RCF(p,q)$ in

Table 4. Entropy values of $EN{T}_f$ for $ZHCF\left(t\right)$.

f	t = 1	t = 2	t = 3	t = 4	t = 5	t = 6	t = 7	t = 8	t = 9	t = 10
$RR$	5.1493	6.5276	7.3359	7.91	8.3554	8.7196	9.0275	9.2943	9.5296	9.7402
$R{M}_2$	5.0718	6.458	7.269	7.8444	8.2908	8.6555	8.9638	9.2309	9.4665	9.6772
J	5.1474	6.5255	7.3338	7.9078	8.3533	8.7174	9.0253	9.2921	9.5274	9.738
$Z_{3,3}$	4.963	6.3634	7.1792	7.757	8.2048	8.5704	8.8794	9.147	9.383	9.594
$ESO$	5.1271	6.5074	7.3164	7.8909	8.3366	8.7009	9.0089	9.2758	9.5112	9.7218

,

Table 5. Entropy values of $EN{T}_f$ for $AHCF\left(t\right)$.

f	t = 1	t = 2	t = 3	t = 4	t = 5	t = 6	t = 7	t = 8	t = 9	t = 10
$RR$	5.1493	7.0861	8.082	8.7473	9.2466	9.6462	9.9792	10.2647	10.5146	10.7366
$R{M}_2$	5.0718	7.0176	8.0163	8.6828	9.1829	9.5829	9.9162	10.202	10.452	10.6742
J	5.1474	7.084	8.0798	8.7451	9.2444	9.644	9.977	10.2625	10.5123	10.7344
$Z_{3,3}$	4.963	6.9251	7.9284	8.5971	9.0983	9.4991	9.833	10.1191	10.3694	10.5919
$ESO$	5.1271	7.0662	8.0629	8.7285	9.228	9.6277	9.9608	10.2464	10.4962	10.7184

, and

Table 6. Entropy values of $EN{T}_f$ for $RCF(p,3)$.

f	t = 1	t = 2	t = 3	t = 4	t = 5	t = 6	t = 7	t = 8	t = 9	t = 10
$RR$	6.2443	6.7513	7.0861	7.3365	7.5366	7.7032	7.846	7.971	8.082	8.1819
$R{M}_2$	6.1703	6.6811	7.0176	7.269	7.4697	7.6367	7.7799	7.905	8.0163	8.1164
J	6.2424	6.7492	7.084	7.3344	7.5344	7.7011	7.8439	7.9688	8.0798	8.1798
$Z_{3,3}$	6.0678	6.5856	6.9251	7.1781	7.3798	7.5476	7.6913	7.8169	7.9284	8.0288
$ESO$	6.223	6.7309	7.0662	7.3169	7.5171	7.6839	7.8268	7.9518	8.0629	8.1628

and Figure 9, respectively.

5. Conclusions

In this paper, we investigate three types of coronene fractals, namely, zigzag hexagonal coronene fractals, armchair hexagonal coronene fractals, and rectangular coronene fractals. We compute the reciprocal Randić index, reduced second Zagreb index, Balaban index, (a, b)-Zagreb index, elliptic Sombor index, and degree-based entropies associated with these indices. The acquired results are valuable for anticipating numerous molecular features of chemical substances, such as boiling point, mean radius, acentric factor, pharmaceutical configuration, and many more concepts. We anticipate that the computed entropy values will provide insights into the structural issues of these fractal compounds and facilitate further research on the properties and behaviors of these fractal frameworks.