Degree-Based Entropy for a Non-Kekulean Benzenoid Graph

Abstract

Tessellations of kekulenes and cycloarenes have a lot of potential as nanomolecular belts for trapping and transporting heavy metal ions and chloride ions because they have the best electronic properties and pore sizes. The aromaticity, superaromaticity, chirality, and novel electrical and magnetic properties of a class of cycloarenes known as kekulenes have been the subject of several experimental and theoretical studies. Through topological computations of superaromatic structures with pores, we investigate the entropies and topological characterization of different tessellations of kekulenes. Using topological indices, the biological activity of the underlying structure is linked to its physical properties in (QSPR/QSAR) research. There is a wide range of topological indices accessible, including degree-based indices, which are used in this work. With the total $\pi$-electron energy, these indices have a lot of iteration. In addition, we use graph entropies to determine the structural information of a non-Kekulean benzenoid graph. In this article, we study the crystal structure of non-Kekulean benzenoid graph ${\mathcal{K}}_n$ and then calculate some entropies by using the degree-based topological indices. We also investigate the relationship between degree-based topological indices and degree-based entropies. This relationship is very helpful for chemist to study the physicochemical characterization of non-Kekulean benzenoid chemical. These numerical values correlate with structural facts and chemical reactivity, biological activities, and physical properties.

This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. Introduction

Chemical graph theory is used to mathematically model molecules in order to review their physical properties. It is also a good idea to characterize chemical structures. Chemical graph theory could be a mathematical branch that combines graph theory and chemistry.

Topological indices are molecular descriptors that can be used to describe these characteristics and specific chemical graphs [1]. The topological index of a chemical composition is a numerical value or continuation of a given structure under consideration that indicates chemical, physical, and biological properties of a chemical molecule structure [2].

It also belongs to a category of nontrivial chemical graph theory applications for exact molecular problem solutions. This theory is essential in the field of chemical sciences and chemical graph theory. More information is on quantity structure activity relationship (QSAR) and quantity structure property relationship (QSPR), which are used to predict biobiota and physicochemical properties in chemical compounds [3, 4].

In this article, $G$ is the connected simple chemical structure, with $VG$ vertices set and $EG$ edges set. Degree of any vertex $u$ is denoted by $\tilde{\Re }u$. The edge between vertices $u$ and $v$ is denoted by $uv$. The total number of atoms linked to $v_j$ of $G$ is denoted by $d_{v_j}$, and it is the atom-bond of every atom of $G$. If $G$ is a graph which contains $m$ atoms and $n$ atom-bonds, then order of $G$, denoted by $G$, is $m$ and the size of $G$ denoted by $SG$ is $n$. An alternating sequence of atoms and atom-bonds in a graph $G$ is known as a path in $G$. If there exists a path between every two atoms in $G$, then $G$ is said to be a connected graph. A $u_i\sim v_j$ geodesic is the shortest path between two atoms $u_i$ and $v_j$ in a connected graph $G$. The number of atom-bonds (length) in a $u_i\sim v_j$ geodesic is called the distance between $u_i$ and $v_j$, denoted by $d{u}_i,v_j$ for $u_i,v_j\in V_G$, in a connected graph [5]. In this article, we construct the non-Kekulean benzenoid graph ${\mathcal{K}}_n$ and computed the $1^{\text{st}}$ redefined Zagreb entropy, $2^{\text{nd}}$ redefined Zagreb entropy, $3^{\text{rd}}$ redefined Zagreb entropy, $4^{\text{th}}$ atom-bond connectivity entropy, $5^{\text{th}}$ geometry arithmetic entropy, and Sanskurti entropy by using their indices. We used the concept of entropy from Shazia Manzoor’ article [6]. The huge amount of infirmations were missing for non-Kekulean benzenoid structure that we find in this paper. This information is much helpful for the chemists to study the physio-chemical properties of the non-Kekulean benzenoid structure.

2. Literature Review

In 2013, Ranjini et al. [7] introduced redefined version of Zagreb indices $ReZ{G}_1$, $ReZ{G}_2$, and $ReZ{G}_3$. These are formulated as$$\begin{array}{}\text{(1)}& \mathrm{Re}Z{G}_1={\displaystyle \sum _{u_i{v}_j\in E_G}\frac{d_{u_i}+d_{v_j}}{d_{u_i}\times d_{v_j}}},\text{(2)}& \mathrm{Re}Z{G}_2={\displaystyle \sum _{u_i{v}_j\in E_G}\frac{d_{u_i}\times d_{v_j}}{d_{u_i}+d_{v_j}}},\text{(3)}& \mathrm{Re}Z{G}_3={\displaystyle \sum _{u_i{v}_j\in E_G}{d}_{u_i}\times d_{v_j}{d}_{u_i}+d_{v_j}}.\end{array}$$

In 2010, Ghorbani and Hosseinzadeh [8] introduced the fourth version of the atom-bond connectivity index $AB{C}_4$ of a graph $G$ and formulated as$$\begin{array}{}\text{(4)}& AB{C}_4={\displaystyle \sum _{u_i{v}_j\in E_G}\sqrt{\frac{S_{u_i}+S_{v_j}-2}{S_{u_i}\times S_{v_j}}}}.\end{array}$$

In 2011, Graovac et al. [9] introduced the fifth version of the geometric arithmetic index GA5 of a graph G as$$\begin{array}{}\text{(5)}& G{A}_{5}G={\displaystyle \sum _{u_i{v}_j\in E_G}\frac{2\sqrt{S_{u_i}\times S_{v_j}}}{S_{u_i}+S_{v_j}}}.\end{array}$$

In 2017, Hosamani introduced the Sanskruti index [10] $S_G$ for a molecular graph $G$ as follows and have worked on it till now in 2021 [11], denoted by S(G),$$\begin{array}{}\text{(6)}& SG={\displaystyle \sum _{u_i{v}_j\in E_G}{\frac{S_{u_i}\times S_{v_j}}{S_{u_i}+S_{v_j}-2}}^3},\end{array}$$

Shannon first introduced the idea of entropy in his famous article [12] in 1948. The unpredictability of information content or the uncertainty of a system is measured by the entropy of a probability distribution. Later on, entropy was applied to graphs and chemical networks and it was developed to better understand the structural information in these networks. Graph entropies have recently gained popularity in fields such as biology, chemistry, ecology, and sociology to name a few. Degree of every atom is extremely important; graph theory and network theory have both conducted extensive research on invariants, which are used as information functionals in science and have been around for a long time. In the following paragraphs, we will go over graph entropy measures that have been used to investigate biological and chemical networks in chronological order [13–15].

In this article, we construct the non-Kekulean benzenoid graph ${\mathcal{K}}_n$ and computed the $1^{\text{st}}$ redefined Zagreb entropy, $2^{\text{nd}}$ redefined Zagreb entropy, $3^{\text{rd}}$ redefined Zagreb entropy, $4^{\text{th}}$ atom-bond connectivity entropy, $5^{\text{th}}$ geometry arithmetic entropy, and Sanskurti entropy by using their indices. We used the concept of entropy from Shazia Manzoor’ article [15].

3. Applications of Entropy

In information theory, the graph entropy is a crucial quantity. It analyses chemical graphs and complex networks for structural information. Distance-based entropy is playing an important role in various forms including different problems in math, biology, chemical graph theory, organic chemistry. The graph is inserted with a topological index by Shannon’s entropy concept and topological indices as molecular descriptors are important tools in (QSAR)/(QSPR) study. Shannon’s seminal work [16] was published in 1948, marking the beginning of modern information theory. Information theory was widely used in biology and chemistry after its early applications in linguistics and electrical engineering (see, for example, in 1953, [17]). Shannon’s entropy formulas [16] were used to figure out a network’s structural information content in 2004 [18].

The work of Rashevsky in 1955 [19] and Trucco in 1956 [20] is closely related to these applications. The following sections go over graph entropy measures that have been used to study biological and chemical networks in chronological order. Entropy measures for graphs have also been widely used in biology, computer science, and structural chemistry (for example, in 2011, see [21]). Entopic network measures have a wide range of applications, ranging from quantitative structure characterization in structural chemistry to exploring biological or chemical properties of molecular graphs in general. We stress that the aforementioned applications are intended to solve a fundamental data analysis problem, such as clustering or classification. We hypothesise that the degree-based entropy introduced in this paper can be used to assess non-Kekulean benzenoid graph.

4. Degree-Based Entropy

In 2014, Chen et al. [22] proposed the definition of entropy of an edge-weighted graph $G$. The $G=V_G,E_G,\mathrm{\Psi }{u}_i{v}_j$, for an edge-weighted graph, where $V_G$, $E_G$, and $\mathrm{\Psi }{u}_i{v}_j$ exemplify the set of vertices, the edge set, and the edge-weight of edge $u_i{v}_j$, respectively. The entropy of edge-weighted graph is defined as$$\begin{array}{}\text{(7)}& EN{T}_{\Psi G}={\displaystyle \sum _{u_i,v_j\in E_G}\frac{\Psi u_i{v}_j}{{\displaystyle \sum _{u_i,v_j\in E_G}\Psi }{u}_i{v}_j}}\log \frac{\Psi u_i{v}_j}{{\displaystyle \sum _{u_i,v_j\in E_G}\Psi }{u}_i{v}_j}.\end{array}$$

By the help of equation 7, other entropies were found [6] and mathematically denoted as follows:

(i) First redefined Zagreb entropy:

  Let $\mathrm{\Psi }{u}_i{v}_j=d_u+d_{v_j}/d_u{d}_{v_j}$. Then, the first redefined Zagreb index (1) is given by$$\begin{array}{}\text{(8)}& \mathrm{Re}Z{G}_1={\displaystyle \sum _{u_i,v_j\in E_G}\frac{d_{u_i}+d_{v_j}}{d_{u_i}{d}_{v_j}}}={\displaystyle \sum _{u_i,v_j\in E_G}\Psi }{u}_i{v}_j.\end{array}$$

  Now, by using these values in (7), the first redefined Zagreb entropy is$$\begin{array}{}\text{(9)}& EN{T}_{\mathrm{Re}Z{G}_1}=\log \mathrm{Re}Z{G}_1-\frac{1}{\mathrm{Re}Z{G}_1}\log {\displaystyle \prod _{u_i,v_j\in E_G}{\frac{d_{u_i}+d_v}{d_{u_i}{d}_{v_j}}}^{d_{u_i}+d_v/d_u{d}_{v_j}}}.\end{array}$$

(ii) Second redefined Zagreb entropy:

  Let $\mathrm{\Psi }{u}_i{v}_j=d_u+d_{v_j}/d_u{d}_{v_j}$. Then, the second redefined Zagreb index (2) is given by$$\begin{array}{}\text{(10)}& ReZ{G}_2={\displaystyle \sum _{u_i,v_j\in E_G}\frac{d_{u_i}{d}_{v_j}}{d_{u_i}+d_{v_j}}}={\displaystyle \sum _{u_i,v_j\in E_G}\Psi }{u}_i{v}_j.\end{array}$$

  Now, by using these values in (7), the second redefined Zagreb entropy is$$\begin{array}{}\text{(11)}& EN{T}_{\mathrm{Re}Z{G}_2}=\log \mathrm{Re}Z{G}_2-\frac{1}{\mathrm{Re}Z{G}_2}\log {\displaystyle \prod _{u_i,v_j\in E_G}{\frac{d_u{d}_{v_j}}{d_{u_i}+d_v}}^{d_{u_i}{d}_{v_j}/d_{u_i}+d_{v_j}}}.\end{array}$$

(iii) Third redefined Zagreb entropy:

  Let $\mathrm{\Psi }{u}_i{v}_j=d_{u_i}{d}_{v_j}{d}_{u_i}+d_{v_j}$. Then, the third redefined Zagreb index (3) is given by$$\begin{array}{}\text{(12)}& ReZ{G}_3={\displaystyle \sum _{u_i,v_j\in E_G}{d}_{u_i}{d}_{v_j}{d}_{u_i}+d_{v_j}}={\displaystyle \sum _{u_i,v_j\in E_G}\Psi }{u}_i{v}_j.\end{array}$$

  Now, by using these values in (7), the third redefined Zagreb entropy is$$\begin{array}{}\text{(13)}& EN{T}_{ReZ{G}_3}=\log ReZ{G}_3-\frac{1}{ReZ{G}_3}\log {{\displaystyle \prod _{u_i,v_j\in E_G}{d}_{u_i}{d}_{v_j}{d}_{u_i}+d_{v_j}}}^{d_{u_i}{d}_{v_j}{d}_{u_i}+d_{v_j}}.\end{array}$$

(iv) Entropy of fourth atom-bond connectivity:

  Let $\mathrm{\Psi }{u}_i{v}_j=\sqrt{S_{u_i}+S_{v_j}-2/S_{u_i}{S}_{v_j}}$. Then, the $4^{\text{th}}$ atom-bond connectivity index (4) is$$\begin{array}{}\text{(14)}& AB{C}_{4}G={\displaystyle \sum _{u_i,v_j\in E_G}\sqrt{\frac{S_{u_i}+S_{v_j}-2}{S_{u_i}{S}_{v_j}}}}={\displaystyle \sum _{u_i,v_j\in E_G}\Psi }{u}_i{v}_j.\end{array}$$

  Here, $S_{u_i}$ is the neighborhood degree sum of vertex $u_i$. Now, by using these values in (7), the third redefined Zagreb entropy is$$\begin{array}{}\text{(15)}& EN{T}_{AB{C}_{4}G}=\log AB{C}_{4}G-\frac{1}{AB{C}_{4}G}\log {{\displaystyle \prod _{u_i,v_j\in E_G}\sqrt{\frac{S_{u_i}+S_{v_j}-2}{S_{u_i}{S}_{v_j}}}}}^{\sqrt{S_{u_i}+S_{v_j}-2/S_{u_i}{S}_{v_j}}}.\end{array}$$

(v) Fifth geometry arithmetic entropy:

  Let $\mathrm{\Psi }{u}_i{v}_j=2\sqrt{S_{u_i}{S}_{v_j}}/S_{u_i}+S_{v_j}$. Then, the fifth geometry arithmetic index (4) is given by$$\begin{array}{}\text{(16)}& G{A}_{5}G={\displaystyle \sum _{u_i,v_j\in E_G}\frac{2\sqrt{S_{u_i}{S}_{v_j}}}{S_{u_i}+S_{v_j}}}={\displaystyle \sum _{u_i,v_j\in E_G}\Psi }{u}_i{v}_j.\end{array}$$

  Now, by using these values in (7), the fifth geometric arithmetic entropy is$$\begin{array}{}\text{(17)}& EN{T}_{G{A}_{5}G}=\log G{A}_{5}G-\frac{1}{G{A}_{5}G}\log {{\displaystyle \prod _{u_i,v_j\in E_G}\frac{2\sqrt{S_{u_i}{S}_{v_j}}}{S_{u_i}+S_{v_j}}}}^{2\sqrt{S_{u_i}{S}_{v_j}}/S_{u_i}+S_{v_j}}.\end{array}$$

(vi) Sanskruti entropy:

  Let $\mathrm{\Psi }{u}_i{v}_j={S_{u_i}\times S_{v_j}/S_{u_i}+S_{v_j}-2}^3$. Then, the Sanskruti index (6) is given by$$\begin{array}{}\text{(18)}& SG={\displaystyle \sum _{u_i,v_j\in E_G}{\frac{S_{u_i}\times S_{v_j}}{S_{u_i}+S_{v_j}-2}}^3}={\displaystyle \sum _{u_i,v_j\in E_G}\Psi }{u}_i{v}_j.\end{array}$$

  Now, by using these values in equation (7), the Sanskruti entropy is$$\begin{array}{}\text{(19)}& EN{T}_{SG}=\log SG-\frac{1}{SG}\log {{\displaystyle \prod _{u_i,v_j\in E_G}{\frac{S_{u_i}\times S_{v_j}}{S_{u_i}+S_{v_j}-2}}^3}}^{{S_{u_i}\times S_{v_j}/S_{u_i}+S_{v_j}-2}^3}.\end{array}$$

  The Kekulean and non-Kekulean structures of benzene are real and distinct due to the presence of rings in the benzenoid form. The specific arrangement of rings in the benzenoid system provides the transformation in series of benzenoid structures of the benzenoid graph that is the way the structures are changed. In the series of concealed non-Kekulean benzenoid graph ${\mathcal{K}}_n$, see [23], where $n$ shows the number of bridges [24] in the center of ${\mathcal{K}}_n$, as shown in Figure 1. Similarly for $n=k$, there are $k$ bridges. Here, in the non-Kekulean benzenoid graph ${\mathcal{K}}_n$, we observed that there are three types of atom-bonds on the basis of valency of every atom. Therefore, by observing this concept of atom-bonds, there are two types of atoms $v_i$ and $v_j$ such that $d_{v_i}=2$ and $d_{v_j}=3$, where $d_{v_i}$ and $d_{v_j}$ mean the valency of atoms $\forall v_i,v_j\in {\mathcal{K}}_n$. The order and size of non-Kekulean benzenoid graphs ${\mathcal{K}}_n$ are$$\begin{array}{c}\text{(20)}& V{\mathcal{K}}_n=26n+7,E{\mathcal{K}}_n=17n+14.\end{array}$$

  Following are the three figures of non-Kekulean benzenoid graphs $K_3$, $K_4$, and $K_5$.

  According to the degree of the atoms, there are three types of atom-bonds in ${\mathcal{K}}_n$: $2\sim 2$, $2\sim 3$, and $3\sim 3$. The atom-bonds partition of ${\mathcal{K}}_n$ is shown as$$\begin{array}{c}\text{(21)}& E_{2\sim 2}=e=u\sim v,\hspace{1em}\forall u,v\in V{\mathcal{K}}_n|d_u=2,d_v=2,E_{2\sim 3}=e=u\sim v,\hspace{1em}\forall u,v\in V{\mathcal{K}}_n|d_u=2,d_v=3,\\ E_{3\sim 3}=e=u\sim v,\hspace{1em}\forall u,v\in V{\mathcal{K}}_n|d_u=3,d_v=3.\end{array}$$

(vii) First redefined Zagreb entropy of ${\mathcal{K}}_n$

  Let $G$ be the non-Kekulean benzenoid graph ${\mathcal{K}}_n$. Then, by using Table 1 in (1), the first redefined Zagreb index is$$\begin{array}{}\text{(22)}& \mathrm{Re}Z{G}_1{\mathcal{K}}_n=26n+7.\end{array}$$

  Now, we are computing the first redefined Zagreb entropy by using Table 1 and (22) in (9) in the following way:$$\begin{array}{c}\text{(23)}& EN{T}_{ReZ{G}_1}{\mathcal{K}}_n=\log 26n+7-\frac{1}{26n+7}\log {{\displaystyle \prod _{E_{2\sim 2}}\frac{d_{u_i}+d_{v_j}}{d_{u_i}{d}_{v_j}}}}^{d_{u_i}+d_{v_j}/d_{u_i}{d}_{v_j}}\times {{\displaystyle \prod _{E_{2\sim 3}}\frac{d_{u_i}+d_{v_j}}{d_{u_i}{d}_{v_j}}}}^{d_{u_i}+d_{v_j}/d_{u_i}{d}_{v_j}}\times {\displaystyle \prod _{E_{3\sim 3}}{\frac{d_{u_i}+d_{v_j}}{d_{u_i}{d}_{v_j}}}^{d_{u_i}+d_{v_j}/d_{u_i}{d}_{v_j}}}=\log 26n+7-\frac{1}{26n+7}\log 8{\frac{4}{4}}^{4/4}\times 4n+3{\frac{5}{6}}^{5/6}\times 13n-6{\frac{6}{9}}^{6/9}.\end{array}$$

  After simplification, in the following equation, we get the actual amount of the first redefined Zagreb entropy.$$\begin{array}{}\text{(24)}& EN{T}_{\mathrm{Re}Z{G}_1}{\mathcal{K}}_n=\log 26n+7-\frac{1}{26n+7}\log 8\times 4n+3{\frac{5}{6}}^{5/6}\times 13n-6{\frac{2}{3}}^{2/3}.\end{array}$$

(viii) Second redefined Zagreb entropy of ${\mathcal{K}}_n$:

  Let $G$ be the non-Kekulean benzenoid graph ${\mathcal{K}}_n$. Then, by using Table 1 in (2), the second redefined Zagreb index is$$\begin{array}{}\text{(25)}& ReZ{G}_2{\mathcal{K}}_n=\frac{243}{10}n+\frac{67}{5}.\end{array}$$

  Now, we are computing the second redefined Zagreb entropy by using Table 1 and (25) in (11) in the following way:$$\begin{array}{c}\text{(26)}& EN{T}_{\mathrm{Re}Z{G}_2}{\mathcal{K}}_n=\log \mathrm{Re}Z{G}_2-\frac{1}{\mathrm{Re}Z{G}_2}\log {\displaystyle \prod _{E2\sim 2}{\frac{d_{u_i}{d}_{v_j}}{d_{u_i}+d_{v_j}}}^{d_{u_i}{d}_{v_j}/d_{u_i}+d_{v_j}}}\times {\displaystyle \prod _{E_{2\sim 3}}{\frac{d_{u_i}{d}_{v_j}}{d_{u_i}+d_{v_j}}}^{d_{u_i}{d}_{v_j}/d_{u_i}+d_{v_j}}}\times {\displaystyle \prod _{E_{3\sim 3}}{\frac{d_{u_i}{d}_{v_j}}{d_{u_i}+d_{v_j}}}^{d_{u_i}{d}_{v_j}/d_{u_i}+d_{v_j}}}=\log \frac{243}{10}n+\frac{67}{5}-\frac{1}{243/10n+67/5}\log 8{\frac{4}{4}}^{4/4}\times 4n+3{\frac{6}{5}}^{6/5}\times 13n-6{\frac{9}{6}}^{9/6}.\end{array}$$

  After simplification, we get the actual amount of second redefined Zagreb entropy in the following equation:$$\begin{array}{}\text{(27)}& EN{T}_{ReZ{G}_2}{\mathcal{K}}_n=\log \frac{1215n+670}{50}-\frac{50}{1215n+670}\log 8\times 4n+3{\frac{6}{5}}^{6/5}\times 13n-6{\frac{3}{2}}^{3/2}.\end{array}$$

(ix) Third redefined Zagreb entropy of ${\mathcal{K}}_n$:

  Let $G$ be the non-Kekulean benzenoid graph ${\mathcal{K}}_n$, then by using Table 1 in (3), the third redefined Zagreb index is$$\begin{array}{}\text{(28)}& ReZ{G}_3{\mathcal{K}}_n=822n+164.\end{array}$$

  Now, we are computing the third redefined Zagreb entropy by using Table 1 and (28) in (13) in the following way:$$\begin{array}{c}\text{(29)}& EN{T}_{ReZ{G}_3}{\mathcal{K}}_n=\log ReZ{G}_3-\frac{1}{ReZ{G}_3}\log {\displaystyle \prod _{E_{2\sim 2}}{d_{u_i}{d}_{v_j}{d}_{u_i}+d_{v_j}}^{d_{u_i}{d}_{v_j}{d}_{u_i}+d_{v_j}}\times {\displaystyle \prod _{E_{2\sim 3}}{d_{u_i}{d}_{v_j}{d}_{u_i}+d_{v_j}}^{d_{u_i}{d}_{v_j}{d}_{u_i}+d_{v_j}}\times {\displaystyle \prod _{E_{3\sim 3}}{d_{u_i}{d}_{v_j}{d}_{u_i}+d_{v_j}}^{d_{u_i}{d}_{v_j}{d}_{u_i}+d_{v_j}}}}}=\log 822n+164-\frac{1}{822n+164}\log 8{16}^{16}\times 4n+3{30}^{30}\times 13n-6{54}^{54}.\end{array}$$

  The above (29) is the actual amount of third redefined Zagreb entropy.

(x) Fourth atom-bond connectivity entropy of ${\mathcal{K}}_n$:

  Table 2 shows the atom-bond-based partition of non-Kekulean graph ${\mathcal{K}}_n$, based on valency sum of end atoms of each degree.

  Let ${\mathcal{K}}_n$ be a non-Kekulean benzenoid graph. Then, by using Table 2 in (4), the forth atom-bond connectivity index is$$\begin{array}{}\text{(30)}& AB{C}_4{\mathcal{K}}_n=\frac{44}{9}+\frac{2\sqrt{2}}{3}+4\sqrt{\frac{11}{42}}n+2\sqrt{\frac{7}{5}}+8\sqrt{\frac{2}{7}}+2\sqrt{\frac{5}{6}}+\frac{4\sqrt{2}}{3}+\frac{\sqrt{14}}{4}-\frac{46}{7}.\end{array}$$

  Now, we are computing the fourth atom-bond connectivity entropy by using Table 2 and (30) in (15) in the following way:$$\begin{array}{c}\text{(31)}& EN{T}_{AB{C}_4}{\mathcal{K}}_n=\log AB{C}_{4}G-\frac{1}{AB{C}_{4}G}\log {\displaystyle \prod _{S_{\mathrm{4,5}}}{\sqrt{\frac{S_{u_i}+S_{v_j}-2}{S_{u_i}{S}_{v_j}}}}^{\sqrt{S_{u_i}+S_{v_j}-2/S_{u_i}{S}_{v_j}}}}\times {\displaystyle \prod _{S_{\mathrm{5,7}}}{\sqrt{\frac{S_{u_i}+S_{v_j}-2}{S_{u_i}{S}_{v_j}}}}^{\sqrt{S_{u_i}+S_{v_j}-2/S_{u_i}{S}_{v_j}}}}\times {\displaystyle \prod _{S_{\mathrm{6,7}}}{\sqrt{\frac{S_{u_i}+S_{v_j}-2}{S_{u_i}{S}_{v_j}}}}^{\sqrt{S_{u_i}+S_{v_j}-2/S_{u_i}{S}_{v_j}}}}\\ \times {\displaystyle \prod _{S_{\mathrm{6,8}}}{\sqrt{\frac{S_{u_i}+S_{v_j}-2}{S_{u_i}{S}_{v_j}}}}^{\sqrt{S_{u_i}+S_{v_j}-2/S_{u_i}{S}_{v_j}}}}\times {\displaystyle \prod _{S_{\mathrm{7,9}}}{\sqrt{\frac{S_{u_i}+S_{v_j}-2}{S_{u_i}{S}_{v_j}}}}^{\sqrt{S_{u_i}+S_{v_j}-2/S_{u_i}{S}_{v_j}}}}\\ \times {\displaystyle \prod _{S_{\mathrm{8,8}}}{\sqrt{\frac{S_{u_i}+S_{v_j}-2}{S_{u_i}{S}_{v_j}}}}^{\sqrt{S_{u_i}+S_{v_j}-2/S_{u_i}{S}_{v_j}}}}\times {\displaystyle \prod _{S_{\mathrm{8,9}}}{\sqrt{\frac{S_{u_i}+S_{v_j}-2}{S_{u_i}{S}_{v_j}}}}^{\sqrt{S_{u_i}+S_{v_j}-2/S_{u_i}{S}_{v_j}}}}\\ \times {\displaystyle \prod _{S_{\mathrm{9,9}}}{\sqrt{\frac{S_{u_i}+S_{v_j}-2}{S_{u_i}{S}_{v_j}}}}^{\sqrt{S_{u_i}+S_{v_j}-2/S_{u_i}{S}_{v_j}}}}.\end{array}$$

  After simplification, we get the exact value of forth atom-bond connectivity entropy$$\begin{array}{c}\text{(32)}& EN{T}_{AB{C}_4}{\mathcal{K}}_n=\log \frac{44}{9}+\frac{2\sqrt{2}}{3}+4\sqrt{\frac{11}{42}}n+2\sqrt{\frac{7}{5}}+8\sqrt{\frac{2}{7}}+2\sqrt{\frac{5}{6}}+\frac{4\sqrt{2}}{3}+\frac{\sqrt{14}}{4}-\frac{46}{7}-\frac{1}{44/9+2\sqrt{2}/3+4\sqrt{11/42}n+2\sqrt{7/5}+8\sqrt{2/7}+2\sqrt{5/6}+4\sqrt{2}/3+\sqrt{14}/4-46/7}\log 8{\sqrt{\frac{7}{20}}}^{\sqrt{7/20}}\times 8{\sqrt{\frac{2}{7}}}^{\sqrt{2/7}}\times 4n{\sqrt{\frac{11}{42}}}^{\sqrt{11/42}}\times 4{\frac{1}{2}}^{1/2}\times 2n+4{\sqrt{\frac{14}{63}}}^{\sqrt{14/63}}\times 2{\sqrt{\frac{7}{32}}}^{\sqrt{7/32}}\times 4{\sqrt{\frac{5}{24}}}^{\sqrt{5/24}}\times 11n-16{\frac{4}{9}}^{4/9}.\end{array}$$

(xi) Fifth geometry arithmetic entropy of ${\mathcal{K}}_n$:

  Let ${\mathcal{K}}_n$ be a non-Kekulean benzenoid graph, then by using Table 2 in (4), the fifth geometry arithmetic index is$$\begin{array}{}\text{(33)}& G{A}_A{\mathcal{K}}_n=11n+\frac{8\sqrt{42}}{13}n+\frac{3\sqrt{7}}{4}n+\frac{3\sqrt{7}}{2}+\frac{4\sqrt{35}}{3}+\frac{32\sqrt{5}}{9}+\frac{16\sqrt{3}}{7}+\frac{48\sqrt{2}}{17}-14.\end{array}$$

  Now, we are computing the fifth geometry arithmetic entropy of ${\mathcal{K}}_n$ by using (33) and Table 2 in (17) in the following way:$$\begin{array}{c}\text{(34)}& EN{T}_{G{A}_5}{\mathcal{K}}_n=\log G{A}_{5}G-\frac{1}{G{A}_{5}G}\log {\displaystyle \prod _{S_{\mathrm{4,5}}}{\frac{2\sqrt{S_{u_i}{S}_{v_j}}}{S_{u_i}+S_{v_j}}}^{2\sqrt{S_{u_i}{S}_{v_j}}/S_{u_i}+S_{v_j}}}\times {\displaystyle \prod _{S_{\mathrm{5,7}}}{\frac{2\sqrt{S_{u_i}{S}_{v_j}}}{S_{u_i}+S_{v_j}}}^{2\sqrt{S_{u_i}{S}_{v_j}}/S_{u_i}+S_{v_j}}}\times {\displaystyle \prod _{S_{\mathrm{6,7}}}{\frac{2\sqrt{S_{u_i}{S}_{v_j}}}{S_{u_i}+S_{v_j}}}^{2\sqrt{S_{u_i}{S}_{v_j}}/S_{u_i}+S_{v_j}}}\\ \times {\displaystyle \prod _{S_{\mathrm{6,8}}}{\frac{2\sqrt{S_{u_i}{S}_{v_j}}}{S_{u_i}+S_{v_j}}}^{2\sqrt{S_{u_i}{S}_{v_j}}/S_{u_i}+S_{v_j}}}\times {\displaystyle \prod _{S_{\mathrm{7,9}}}{\frac{2\sqrt{S_{u_i}{S}_{v_j}}}{S_{u_i}+S_{v_j}}}^{2\sqrt{S_{u_i}{S}_{v_j}}/S_{u_i}+S_{v_j}}}\\ \times {\displaystyle \prod _{S_{\mathrm{8,8}}}{\frac{2\sqrt{S_{u_i}{S}_{v_j}}}{S_{u_i}+S_{v_j}}}^{2\sqrt{S_{u_i}{S}_{v_j}}/S_{u_i}+S_{v_j}}}\times {\displaystyle \prod _{S_{\mathrm{8,9}}}{\frac{2\sqrt{S_{u_i}{S}_{v_j}}}{S_{u_i}+S_{v_j}}}^{2\sqrt{S_{u_i}{S}_{v_j}}/S_{u_i}+S_{v_j}}}\\ \times {\displaystyle \prod _{S_{\mathrm{9,9}}}{\frac{2\sqrt{S_{u_i}{S}_{v_j}}}{S_{u_i}+S_{v_j}}}^{2\sqrt{S_{u_i}{S}_{v_j}}/S_{u_i}+S_{v_j}}}=\log G{A}_{5}G-\frac{1}{G{A}_{5}G}\log 8{\frac{2\sqrt{20}}{9}}^{2\sqrt{20}/9}\times 8{\frac{2\sqrt{35}}{12}}^{2\sqrt{35}/12}\\ \times 4n.{\frac{2\sqrt{42}}{13}}^{2\sqrt{42}/13}\times 4{\frac{2\sqrt{48}}{14}}^{2\sqrt{48}/14}\times 2n+2{\frac{2\sqrt{63}}{16}}^{2\sqrt{63}/16}\times 2{\frac{2\sqrt{64}}{16}}^{2\sqrt{64}/16}\\ \times 4{\frac{2\sqrt{72}}{17}}^{2\sqrt{72}/17}\times 11n-16{\frac{2\sqrt{81}}{18}}^{2\sqrt{81}/18}.\end{array}$$

  By using the value of fifth geometry arithmetic index in the above expiration, we get the exact value of fifth geometry arithmetic entropy of ${\mathcal{K}}_n$:$$\begin{array}{c}\text{(35)}& EN{T}_{G{A}_5}=\log 11n+\frac{8\sqrt{42}}{13}n+\frac{3\sqrt{7}}{4}n+\frac{3\sqrt{7}}{2}+\frac{4\sqrt{35}}{3}+\frac{32\sqrt{5}}{9}+\frac{16\sqrt{3}}{7}+\frac{48\sqrt{2}}{17}-14/-11n+\frac{8\sqrt{42}}{13}n+\frac{3\sqrt{7}}{4}n+\frac{3\sqrt{7}}{2}+\frac{4\sqrt{35}}{3}+\frac{32\sqrt{5}}{9}+\frac{16\sqrt{3}}{7}+\frac{48\sqrt{2}}{17}-14\times \log 8{\frac{4\sqrt{5}}{9}}^{4\sqrt{5}/9}\times 8{\frac{\sqrt{35}}{6}}^{\sqrt{35}/6}\times 4n{\frac{2\sqrt{42}}{13}}^{2\sqrt{42}/13}\times 4{\frac{4\sqrt{3}}{7}}^{4\sqrt{3}/7}\\ \times 2n+2{\frac{3\sqrt{7}}{8}}^{3\sqrt{7}/8}\times 2\times 4{\frac{12\sqrt{2}}{17}}^{12\sqrt{2}/17}\times 11n-16.\end{array}$$

(xii) Sanskruti entropy of ${\mathcal{K}}_n$:

  Let ${\mathcal{K}}_n$ be a non-Kekulean benzenoid graph. Then, by using Table 2 in (6), Sanskruti index is$$\begin{array}{}\text{(36)}& S{\mathcal{K}}_n=\frac{54759n-29231}{100}.\end{array}$$

  Now, we are computing Sanskruti entropy by using (36) and Table 2 in (17) in the following way:$$\begin{array}{c}\text{(37)}& EN{T}_S{\mathcal{K}}_n=\log S{\mathcal{K}}_n-\frac{1}{S{\mathcal{K}}_n}\log {\displaystyle \prod _{S_{\mathrm{4,5}}}{\frac{S_{u_i}\times S_{v_j}}{S_{u_i}+S_{v_j}-2}}^{3{S_{u_i}\times S_{v_j}/S_{u_i}+S_{v_j}-2}^3}}\times {\displaystyle \prod _{S_{\mathrm{5,7}}}{\frac{S_{u_i}\times S_{v_j}}{S_{u_i}+S_{v_j}-2}}^{3{S_{u_i}\times S_{v_j}/S_{u_i}+S_{v_j}-2}^3}}\times {\displaystyle \prod _{S_{\mathrm{6,7}}}{\frac{S_{u_i}\times S_{v_j}}{S_{u_i}+S_{v_j}-2}}^{3{S_{u_i}\times S_{v_j}/S_{u_i}+S_{v_j}-2}^3}}\\ \times {\displaystyle \prod _{S_{\mathrm{6,8}}}{\frac{S_{u_i}\times S_{v_j}}{S_{u_i}+S_{v_j}-2}}^{3{S_{u_i}\times S_{v_j}/S_{u_i}+S_{v_j}-2}^3}}\times {\displaystyle \prod _{S_{\mathrm{7,9}}}{\frac{S_{u_i}\times S_{v_j}}{S_{u_i}+S_{v_j}-2}}^{3{S_{u_i}\times S_{v_j}/S_{u_i}+S_{v_j}-2}^3}}\\ \times {\displaystyle \prod _{S_{\mathrm{8,8}}}{\frac{S_{u_i}\times S_{v_j}}{S_{u_i}+S_{v_j}-2}}^{3{S_{u_i}\times S_{v_j}/S_{u_i}+S_{v_j}-2}^3}}\times {\displaystyle \prod _{S_{\mathrm{8,9}}}{\frac{S_{u_i}\times S_{v_j}}{S_{u_i}+S_{v_j}-2}}^{3{S_{u_i}\times S_{v_j}/S_{u_i}+S_{v_j}-2}^3}}\\ \times {\displaystyle \prod _{S_{\mathrm{9,9}}}{\frac{S_{u_i}\times S_{v_j}}{S_{u_i}+S_{v_j}-2}}^{3{S_{u_i}\times S_{v_j}/S_{u_i}+S_{v_j}-2}^3}}.\end{array}$$

  We get the actual amount of Sanskruti entropy of ${\mathcal{K}}_n$ by using the value of Sanskruti index in the above expiration.$$\begin{array}{c}\text{(38)}& EN{T}_S{\mathcal{K}}_n=\log \frac{54759n-29231}{100}-\frac{100}{54759n-29231}\log 8{\frac{8000}{343}}^{8000/343}\times 8{\frac{343}{8}}^{343/8}\times 4n{\frac{74088}{1331}}^{74088/1331}\times 4{64}^{64}\\ \times 2n+2{\frac{729}{8}}^{729/8}\times 2{\frac{32768}{343}}^{32768/343}\times 4{\frac{13824}{125}}^{13824/125}\times 11n-16{\frac{531441}{4096}}^{531441/4096}.\end{array}$$

[figure(s) omitted; refer to PDF]

Table 1

Atom-bonds-based partition of each atom of ${\mathcal{K}}_n$.

Types of atom-bonds	$E_{2\sim 2}$	$E_{2\sim 3}$	$E_{3\sim 3}$
Frequency of atom-bonds	8	$4n+3$	$13n-6$

Table 2

Edge partition (sum of valency of the neighborhood atoms).

$S_{u_i},S_{v_j}$	Number of atom-bonds
$S_{\mathrm{4,5}}$	8
$S_{\mathrm{5,7}}$	8
$S_{\mathrm{6,7}}$	$4n$
$S_{\mathrm{6,8}}$	4
$S_{\mathrm{7,9}}$	$2n+4$
$S_{\mathrm{8,8}}$	2
$S_{\mathrm{8,9}}$	4
$S_{\mathrm{9,9}}$	$11n-16$

5. Numerical and Graphical Representation

The numerical representation and the graphical representation are dedicated in Figure 2. We can easily see, from Figure 2, that all indices are in increasing order as the value of $n$ is increasing. Comparison of entropy for $ENTAB{C}_4{\mathcal{K}}_n$ and $ENTG{A}_5{\mathcal{K}}_n$ is taken for $n>1$.

[figure(s) omitted; refer to PDF]

6. Conclusion

For the construction of entropy-based measures to characterize the structure of complex networks, many graph invariants have been used. We study graph entropies based on vertex degrees using so-called information functionals, which are based on Shannon’s entropy. There has been very little work done to find the extremal values of Shannon entropy-based graph measures. The main contribution of this paper is to prove some extreme values for the key focus area entropy of certain non-Kekulean benzenoid graph. In this research, we investigate the graph entropies associated with a new information function using Shannon’s entropy and Chen et al’s entropy definitions and evaluate a relationship between degree-based topological indices and degree-based entropies. The degree-based entropies for crystallographic structures of non-Kekulean benzenoid graph ${\mathcal{K}}_n$ were calculated by their indices, which leads us to the physicochemical properties of the non-Kekulean benzenoid graph ${\mathcal{K}}_n$.

In the future, we hope to expand this concept to include various chemical structures with the help of chemists, allowing researchers to pursue new avenues in this field.

Authors’ Contributions

The authors contributed equally in the analysis and write up of the manuscript.

Acknowledgments

The authors are very thankful to the Jahangirnagar University, Savar, Dhaka, Bangladesh, for supporting this work.

References

[1] P. C. S. Costa, J. S. Evangelista, I. Leal, P. C. M. L. Miranda, "Chemical graph theory for property modeling in QSAR and QSPR-charming QSAR & QSPR," Mathematics, vol. 9 no. 1,DOI: 10.3390/math9010060, 2020.

[2] N. Salamat, M. Kamran, S. Ali, M. Alam, R. H. Khan, "Several characterizations on degree-based topological indices for star of david network," Journal of Mathematics, vol. 2021,DOI: 10.1155/2021/9178444, 2021.

[3] S. Ediza, I. iftia, Z. Tasb, M. Cancana, M. Reza, "A Note on QSPR Analysis of Total Zagreb and Total Randic Indices of Octanes," Eurasian Chemical Communications, vol. 3, pp. 139-145, 2021.

[4] L. Fu, Z. Y. Yang, Z. J. Yang, M. Z. Yin, A. P. Lu, X. Chen, S. Liu, T. J. Hou, D. S. Cao, "QSAR-assisted-MMPA to expand chemical transformation space for lead optimization," Briefings in Bioinformatics, vol. 22 no. 5,DOI: 10.1093/bib/bbaa374, 2021.

[5] A. Ali, K. C. Das, D. Dimitrov, B. Furtula, "Atombond connectivity index of graphs: a review over extremal results and bounds," Discrete Mathematics Letters, vol. 5 no. 1, pp. 68-93, 2021.

[6] S. Manzoor, M. K. Siddiqui, S. Ahmad, "On entropy measures of molecular graphs using topological indices," Arabian Journal of Chemistry, vol. 13 no. 8, pp. 6285-6298, DOI: 10.1016/j.arabjc.2020.05.021, 2020.

[7] P. S. Ranjini, V. Lokesha, A. Usha, "Relation between phenylene and hexagonal squeeze using harmonic index," International Journal of Graph Theory, vol. 1 no. 4, pp. 116-121, 2013.

[8] S. M. Hosamani, "Computing sanskruti index of nanostructures," Journal Applied Mathematics and Computing, vol. 54, 2016.

[9] A. Graovac, M. Ghorbani, M. A. Hosseinzadeh, "Computing fifth geometric-arithmetic index for nanostar dendrimers," Journal of Mathematical Nanoscience, vol. 1 no. 1-2, pp. 33-42, 2011.

[10] S. M. Hosamani, "Computing Sanskruti index of certain nanostructures," Journal of Applied Mathematics and Computing, vol. 54 no. 1, pp. 425-433, DOI: 10.1007/s12190-016-1016-9, 2017.

[11] M. Javaid, U. Amin, A. Raheem, "Extremal unicyclic graphs with respect to the Sanskruti index," Asian-European Journal of Mathematics, vol. 14 no. 01,DOI: 10.1142/s1793557120501478, 2021.

[12] C. E. Shannon, "A mathematical theory of communication," Bell System Technical Journal, vol. 27 no. 3, pp. 379-423, DOI: 10.1002/j.1538-7305.1948.tb01338.x, 1948.

[13] W. Huang, Y. Zhang, Y. Yu, Y. Xu, M. Xu, R. Zhang, G. J. De Dieu, D. Yin, Z. Liu, "Historical data-driven risk assessment of railway dangerous goods transportation system: comparisons between Entropy Weight Method and Scatter Degree Method," Reliability Engineering & System Safety, vol. 205,DOI: 10.1016/j.ress.2020.107236, 2021.

[14] K. Julietraja, P. Venugopal, S. Prabhu, A. K. Arulmozhi, M. K. Siddiqui, "Structural analysis of three types of PAHs using entropy measures," Polycyclic Aromatic Compounds,DOI: 10.1080/10406638.2021.1884101, 2021.

[15] C.-X. Nie, F.-T. Song, "Entropy of graphs in financial markets," Computational Economics, vol. 57 no. 4, pp. 1149-1166, DOI: 10.1007/s10614-020-10007-3, 2021.

[16] C. E. Shannon, "A mathematical theory of communication," ACM SIGMOBILE Mobile Computing and Communications Review, vol. 5 no. 1,DOI: 10.1145/584091.584093, 2001.

[17] H. J. Morowitz, "Some order-disorder considerations in living systems," Bulletin of Mathematical Biophysics, vol. 17 no. 2, pp. 81-86, DOI: 10.1007/bf02477985, 1955.

[18] R. V. Solé, S. Valverde, "Information theory of complex networks: on evolution and architectural constraints," Complex Networks, pp. 189-207, DOI: 10.1007/978-3-540-44485-5_9, 2004.

[19] N. Rashevsky, "Life, information theory, and topology," Bulletin of Mathematical Biophysics, vol. 17 no. 3, pp. 229-235, DOI: 10.1007/bf02477860, 1955.

[20] E. Trucco, "A note on the information content of graphs," Bulletin of Mathematical Biophysics, vol. 18 no. 2, pp. 129-135, DOI: 10.1007/bf02477836, 1956.

[21] M. Dehmer, A. Mowshowitz, "A history of graph entropy measures," Information Sciences, vol. 181 no. 1, pp. 57-78, DOI: 10.1016/j.ins.2010.08.041, 2011.

[22] Z. Chen, M. Dehmer, Y. Shi, "A note on distance-based graph entropies," Entropy, vol. 16 no. 10, pp. 5416-5427, DOI: 10.3390/e16105416, 2014.

[23] M. Ibrahim, N. Zahra, M. K. Siddiqui, "On ve-degree and ev-degree based topological indices for the series of benzenoid graphs," Polycyclic Aromatic Compounds,DOI: 10.1080/10406638.2021.1926294, 2021.

[24] A. Amrullah, A. Syahrul, M. Turmuzi, K. Baidowi, K. Nani, "The bridge graphs partition dimension," Journal of Physics: Conference Series, vol. 1779 no. 1,DOI: 10.1088/1742-6596/1779/1/012086, 2021.