Article Volume 12, Issue 3, 2022, 2970 - 2987 https://doi.org/10.33263/BRIAC123.29702987

Topological Analysis of PAHs using Irregularity based Indices

Julietraja Konsalraj 1,* , Venugopal Padmanabhan 2 , Chellamani Perumal 3

1 Department of Mathematics, Sri Sivasubramaniya Nadar College of Engineering, Kalavakkam – 603 110, India; [email protected] (J.K.); 2 Department of Mathematics, Shiv Nadar University Chennai, Kalavakkam – 603 110, India; [email protected] (V.P.); 3 Department of Mathematics, Sacred Heart College (Autonomous), Tirupattur – 635 601, Tirupattur Dt., India; [email protected] (C.P.); * Correspondence: [email protected] (J.K.); Scopus Author ID 57218631900 Received: 8.06.2021; Revised: 15.07.2021; Accepted: 22.07.2021; Published: 8.08.2021 Abstract: Topological descriptors are non-empirical graph invariants that characterize the structures of chemical molecules. The structural descriptors are vital components of QSAR/QSPR studies which form the basis for theoretical chemists to design and investigate new chemical structures. Irregularity indices are a class of topological descriptors that have been employed to study certain chemical properties of compounds. This article aims to compute analytical expressions of irregularity indices for three important classes of polycyclic aromatic hydrocarbons. The intriguing properties of these classes of compounds have several potential applications in wide-raging fields, which warrant a study of their properties from a structural perspective. Additionally, the 3D graphical representations of a few indices are presented, which will aid in analyzing the similarity of behavior among the indices.

Keywords: topological descriptors; benzenoid systems; graph-theoretical methods; irregularity indices. © 2021 by the authors. This article is an open-access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

1. Introduction

A Polycyclic Aromatic Hydrocarbon (PAH) is an organic compound that contains at least two aromatic rings placed in linear, angular, or clustered structures. PAHs are primarily produced by anthropogenic and natural combustion processes, with palpable environmental and human health effects. They are also marked by intriguing characteristics such as spectra, specific reactivity, and photophysics [1,2]. PAHs are primarily studied for toxicity analyses as these substances are toxic and carcinogenic and are also present ubiquitously. The recent development in the field of nano-sciences, particularly in graphene-based molecules, has also rekindled the interest among researchers in the study of PAHs [3]. Hexabenzocoronene [4] (HBC) is a PAH of empirical formula C42H18. The molecular structure includes a coronene as its core and additional benzene placed between each pair of rings on the circumference. HBC molecules exhibit outstanding stability and self-assembly. The HBC and their derivatives are applied across wide-ranging technologies from batteries and solar cells to sensors and semiconductors [5,6]. Dodeca-benzo-circumcoronene (DBC) [7] is a PAH that consists of thirty-one benzene rings with the empirical formula C84H24. Twelve of these rings are placed such that they enclose

https://biointerfaceresearch.com/ 2970 https://doi.org/10.33263/BRIAC123.29702987 a circumcoronene molecule at the center. The structure of DBC provides a prototype for graphene nanoflakes. Hexa-cata-hexabenzocoronene (cHBC) is a PAH of empirical formula C48H24. The structure comprises a coronene molecule at its center and an additional benzene ring placed on top of each ring along the perimeter. It is also widely known as contorted hexabenzocoronene due to its twisted structure [8], which is used in several applications such as photovoltaic materials, chemoresponsive, and photoresponsive transistors, and in Li-ion batteries [9,10]. This article focuses on studying these three structures from a structural perspective using irregularity indices. A topological index or a molecular descriptor is a distinct number that quantifies the topology of a molecular structure and its characteristics [11]. The topological descriptors provide an efficient alternative to complex quantum chemical calculations in the theoretical analysis of molecular structures and hence have been studied extensively [12]. Recently, Julietraja and Venugopal computed the degree-based indices for coronoid structures [13]. Chu et al. computed the degree- and irregularity-based molecular descriptors for benzenoid systems [14]. Julietraja et al. also computed the vertex-degree based (VDB) indices using M- polynomial for certain prominent classes of polycyclic aromatic hydrocarbons. For certain types of benzenoid systems, degree-based entropy measures were also studied [15-17]. A topological index is called an irregularity index if the index is non-zero for a non-regular molecular graph and becomes zero for a regular molecular graph [18]. Irregularity indices are particularly applied in the quantitative study of non-regular graphs based on their topology [19]. The irregularity measures of graphs are used in the quantitative structure-activity relationship (QSAR) and quantitative structure-property relationship (QSPR) studies for estimating several chemical and physical characteristics, such as melting and boiling points enthalpy of vaporization, resistance, entropy, and toxicity [20]. The scientific significance of the above structures warrants a study from a structural perspective. Prabhu et al. [21] have analyzed these molecular compounds using distance-based indices. In this article, we attempt to investigate these structures using irregularity indices. A very few articles in the literature have investigated chemical structures using irregularity indices. Hence, this article will add substantial weightage to the current understanding of PAHs.

2. Graph theoretical concepts

In this article, 훤(푉, 퐸) is a connected graph. 푉(훤) represents the vertex set, and 퐸(훤) denotes the edge set of the graph. The degree of a vertex is depicted as 푑푒푔훤(푣) is the number of edges that contain the vertex 푣 as an endpoint [22]. There exists a very extensive literature on degree-based topological descriptors, which show a strong correlation with several physical and chemical properties of PAHs. Some of these indices, as defined below, have been used to compute the irregularity indices. Definition 1. The first and second Zagreb indices of a graph 훤 are defined [23] as

푀1(훤) = ∑ (푑푒푔훤(푥2) + 푑푒푔훤(푦2))

푥2푦2∈퐸

푀2(훤) = ∑ (푑푒푔훤(푥2) ∙ 푑푒푔훤(푦2))

푥2푦2∈퐸

https://biointerfaceresearch.com/ 2971 https://doi.org/10.33263/BRIAC123.29702987 Definition 2. The reciprocal Randić index [24] is described as

푅푅(퐺) = ∑ √(푑푒푔훤(푥2) ∙ 푑푒푔훤(푦2))

푥2푦2∈퐸

Definition 3. The forgotten topological index [25], which is defined as 2 2 퐹(퐺) = ∑ (푑푒푔훤(푥2) + 푑푒푔훤(푦2) )

푥2푦2∈퐸

2.1. Irregularity-based indices for QSPR/QSAR studies.

Table 1. Irregularity-based indices [20].

S. No. Irregularity based indices 2 2 2푚 푀1(훤) 2푚 1. 푉퐴푅(훤) = ∑ (푑푒푔 (푥 ) − ) = − ( ) 훤 2 푛 푛 푛 푥2∈푣 2푚 2푚 2. 퐼푅1(훤 ) = ∑ 푑푒푔 (푥 )3 − ∑ 푑푒푔 (푥 )2 = 퐹(훤) − 푀 (훤) 훤 2 푛 훤 2 푛 1 푥2∈푣 푢∈푣 3. ∑푥 푦 ∈퐸 (푑푒푔훤(푥2) ∙ 푑푒푔훤(푦2)) 2푚 푀 (훤) 2푚 퐼푅2(훤) = √ 2 2 − = √ 2 − 푚 푛 푚 푛 푑푒푔 (푥 ) 푑푒푔 (푦 ) 4. 퐼푅퐷퐼퐹 (훤) = ∑ | 훤 2 − 훤 2 | 푑푒푔훤(푦2) 푑푒푔훤(푦2) 푥2푦2∈퐸

5. 퐴퐿(훤) = ∑ |푑푒푔훤(푥2) − 푑푒푔훤(푦2)|

푥2푦2∈퐸

6. 퐼푅퐿(훤 ) = ∑ |푙푛 푑푒푔훤(푥2) − 푙푛 푑푒푔훤(푦2)|

푥2푦2∈퐸 |푑푒푔 (푥 ) − 푑푒푔 (푦 )| 7. 퐼푅퐿푈( 훤) = ∑ 훤 2 훤 2 min (푑푒푔훤(푥2) ∙ 푑푒푔훤(푦2)) 푥2푦2∈퐸 |푑푒푔 (푥 ) − 푑푒푔 (푦 )| 8. 퐼푅퐿퐹( 훤) = ∑ 훤 2 훤 2 √(푑푒푔훤(푥2) ∙ 푑푒푔훤(푦2)) 푥2푦2∈퐸

2 9. 퐼푅퐹(훤 ) = ∑ (푑푒푔훤(푥2) − 푑푒푔훤(푦2)) = 퐹(훤) − 2푀2(훤)

푥2푦2∈퐸 |푑푒푔 (푥 ) − 푑푒푔 (푦 )| 10. 퐼푅퐿퐴( 훤) = 2 ∑ 훤 2 훤 2 (푑푒푔훤(푥2) + 푑푒푔훤(푦2)) 푥2푦2∈퐸

−1/2 −1/2 2 11. 퐼푅퐴(훤 ) = ∑ (푑푒푔훤(푥2) − 푑푒푔훤(푦2) )

푥2푦2∈퐸

1/2 1/2 2 12. 퐼푅퐵(훤 ) = ∑ (푑푒푔훤(푥2) − 푑푒푔훤(푦2) )

푥2푦2∈퐸

13. ∑푥 푦 ∈퐸 √(푑푒푔훤(푥2) ∙ 푑푒푔훤(푦2)) 2푚 푅푅(훤) 2푚 퐼푅퐶(훤) = 2 2 − = − 푚 푛 푚 푛 (푑푒푔 (푥 ) + 푑푒푔 (푦 )) 14. 퐼푅퐺퐴( 훤) = ∑ 푙푛 ( 훤 2 훤 2 ) 2√(푑푒푔훤(푥2) ∙ 푑푒푔훤(푦2)) 푥2푦2∈퐸 1 15. 퐼푅푅 (훤 ) = ∑ |푑푒푔 (푥 ) − 푑푒푔 (푦 )| 푡 2 훤 2 훤 2 푥2푦2∈퐸

3. Methods

The results presented in this article are calculated by employing the degree counting method, analytical techniques, and edge partition method. The analytical expressions of

https://biointerfaceresearch.com/ 2972 https://doi.org/10.33263/BRIAC123.29702987 irregularity descriptors have been computed using Maple 2016, and the 3D visualization of certain computed indices is also generated using the same tool. ChemDraw Ultra 18.1 is used in representing the molecular structures of the PAH systems visually.

4. Computing the irregularity-based indices for 푯푩푪(풓)

Let Γ be 퐻퐵퐶(푟). The cardinality of 퐻퐵퐶(푟) is 푉(Γ) = 18푟2 − 18푟 + 6 = 푛 and 퐸(Γ) = 27푟2 − 33푟 + 12 = 푚, where 푚 and 푛 indicates the total number of vertices and edges of 퐻퐵퐶(푟) and illustrated in Figure 1.

Table 2. Edge partition Table for 퐻퐵퐶(푟).

(푑푒푔훤(푥2) , 푑푒푔Γ(푦2)) Total number of edges Set of edges 훼 (2,2) 6푟 |퐸{2,2}| (2,3) 12푟 − 12 훽 |퐸{2,3}| (3,3) 27푟2 − 51푟 + 24 훾 |퐸{3,3}|

(i) (ii)

(iii) (iv)

Figure 1. (i) Coronene; (ii) 퐻퐵퐶(2); (iii) 퐻퐵퐶(3); (iv) 퐻퐵퐶(4).

https://biointerfaceresearch.com/ 2973 https://doi.org/10.33263/BRIAC123.29702987 Theorem 1. If Γ is the 퐻퐵퐶(푟), then the irregularity-based indices are computed as 27푟4−90푟3+23푟2+28푟−16 1. 푉퐴푅(훤) = ( ) (3푟2−3푟+1)2 24(9푟3−24푟2+20푟−5) 2. 퐼푅1(훤) = ( ) 3푟2−3푟+1 √54푟2−97푟+47⁄9푟2−11푟+4∙(3푟2−3푟+1) 2−9푟2+11푟−4 3. 퐼푅2(훤) = √ 3푟2−3푟+1 1 4. 퐼푅퐷퐼퐹(훤) = ∙ (12푟 − 12) 3 5. 퐴퐿(훤) = 12푟 − 12 6. 퐼푅퐿(훤) = 0.40545 ∙ (12푟 − 12) 1 7. 퐼푅퐿푈(훤) = ∙ (12푟 − 12) 2 1 8. 퐼푅퐿퐹(훤) = ∙ (12푟 − 12) √6 9. 퐼푅퐹(훤) = 12푟 − 12 2 10. 퐼푅퐿퐴(훤) = ∙ (12푟 − 12) 5 11. 퐼푅퐴(훤) = 0.16832 ∙ (12푟 − 12) 12. 퐼푅퐵(훤) = 0.10106 ∙ (12푟 − 12) (12푟3−24푟2+16푟−4) 6+27푟4−120푟3+167푟2−96푟+22 13. 퐼푅퐶(훤) = ( √ ) 27푟4−60푟3+54푟2−23푟+4 14. 퐼푅퐺퐴(훤) = (0.20391푒 − 1) ∙ (12푛 − 12) 1 15. 퐼푅푅 (훤) = ∙ (12푟 − 12) 푡 2 Proof: Let 훤 be the 퐻퐵퐶(푟) network, then the number of vertices and edges of 퐻퐵퐶(푟) is 푉(훤) = 18푟2 − 18푟 + 6 and 퐸(훤) = 27푟2 − 33푟 + 12. From Table 2, it is noticeable that 훼 |퐸{2,2}| = 6푟

훽 |퐸{2,3}| = 12푟 − 12

훾 2 |퐸{3,3}| = 27푟 − 51푟 + 24.

By using the definitions of 푀1(훤), 푀2(훤), 푅푅(훤) and 퐹(훤),we obtain the following result as 2 푀1(Γ) = 216푟 − 360푟 + 168 2 푀2(Γ) = 324푟 − 582푟 + 282 푅푅(Γ) = 108푟2 + (12푟 − 12)√6 − 210푟 + 114 퐹(Γ) = 648푟2 − 1152푟 + 552 ퟐ풎 ퟐ 푴 (횪) ퟐ풎 ퟐ 푽푨푹(횪) = ∑ (풅풆품 (풙 ) − ) = ퟏ − ( ) 횪 ퟐ 풏 풏 풏 풙ퟐ∈풗 Now put the values of 푀1(Γ) , 푛 and 푚 in 푉퐴푅(Γ) then the result is obtained as 2 216푟2−360푟 2(27푟2−33푟+12) = ( ) − ( ) 18푟2−18푟+6 18푟2−18푟+6 27푟4−90푟3+23푟2+28푟−16 = ( ) (3푟2−3푟+1)2 ퟐ풎 ퟐ풎 푰푹ퟏ(횪) = ∑ 풅풆품 (풙 )ퟑ − ∑ 풅풆품 (풙 )ퟐ = 푭(횪) − 푴 (횪) 횪 ퟐ 풏 횪 ퟐ 풏 ퟏ 풙ퟐ∈풗 풙ퟐ∈풗

https://biointerfaceresearch.com/ 2974 https://doi.org/10.33263/BRIAC123.29702987

Now put the values of 퐹(Γ), 푀1(Γ) , 푛 and 푚 in 퐼푅1(Γ) then the result is obtained as 2(27푟2 − 33푟 + 12) = 648푟2 − 1152푟 + 552 − ( ) ∙ (216푟2 − 360푟 + 168) 18푟2 − 18푟 + 6 24(9푟3 − 24푟2 + 20푟 − 5) = ( ) 3푟2 − 3푟 + 1

∑풙 풚 ∈푬 (풅풆품횪(풙ퟐ) ∙ 풅풆품횪(풚ퟐ)) ퟐ풎 푴 (횪) ퟐ풎 푰푹ퟐ(횪) = √ ퟐ ퟐ − = √ ퟐ − 풎 풏 풎 풏

324푟2−582푟+282 2(27푟2−33푟+12) = (√ ) − ( ) 27푟2−33푟+12 18푟2−18푟+6

√54푟2 − 97푟 + 47⁄9푟2 − 11푟 + 4 ∙ (3푟2 − 3푟 + 1)√2 − 9푟2 + 11푟 − 4 = 3푟2 − 3푟 + 1

풅풆품 (풙 ) 풅풆품 (풚 ) 푰푹푫푰푭(휞) = ∑ | 휞 ퟐ − 휞 ퟐ | 풅풆품휞(풚ퟐ) 풅풆품휞(풚ퟐ) 풙ퟐ풚ퟐ∈푬 2 2 2 3 3 3 = | − | ∙ |퐸훼 | + | − | ∙ |퐸훽 | + | − | ∙ |퐸훾 | 2 2 {2,2} 3 3 {2,3} 3 3 {3,3} 1 = ∙ (12푟 − 12) 3 ( ) ∑ | ( ) ( )| 푨푳 횪 = 풙ퟐ풚ퟐ∈푬 풅풆품횪 풙ퟐ − 풅풆품횪 풚ퟐ 훼 훽 훾 = |2 − 2| ∙ |퐸{2,2}| + |2 − 3| ∙ |퐸{2,3}| + |3 − 3| ∙ |퐸{3,3}| = 12푟 − 12 ( ) ∑ | ( ) ( )| 푰푹푳 휞 = 풙ퟐ풚ퟐ∈푬 풍풏 풅풆품휞 풙ퟐ − 풍풏 풅풆품휞 풚ퟐ 훼 훽 훾 = |푙푛2 − 푙푛2| ∙ |퐸{2,2}| + |푙푛2 − 푙푛3| ∙ |퐸{2,3}| + |푙푛3 − 푙푛3| ∙ |퐸{3,3}| = 0.40545 ∙ (12푟 − 12) ( ) ∑ |풅풆품휞(풙ퟐ)−풅풆품휞(풚ퟐ)| 푰푹푳푼 휞 = 풙ퟐ풚ퟐ∈푬 풎풊풏(풅풆품휞(풙ퟐ)∙풅풆품휞(풚ퟐ)) |2−2| |2−3| |3−3| = ∙ |퐸훼 | + ∙ |퐸훽 | + ∙ |퐸훾 | 2 {2,2} 2 {2,3} 3 {3,3} 1 = ∙ (12푟 − 12) 2 ( ) ∑ |풅풆품휞(풙ퟐ)−풅풆품휞(풚ퟐ)| 푰푹푳푭 휞 = 풙ퟐ풚ퟐ∈푬 √(풅풆품휞(풙ퟐ)∙풅풆품휞(풚ퟐ)) |2−2| |2−3| |3−3| = ∙ |퐸훼 | + ∙ |퐸훽 | + ∙ |퐸훾 | √4 {2,2} √6 {2,3} √9 {3,3} 1 = ∙ (12푟 − 12) √6 ( ) ∑ ( ( ) ( ))ퟐ ( ) ( ) 푰푹푭 휞 = 풙ퟐ풚ퟐ∈푬 풅풆품휞 풙ퟐ − 풅풆품휞 풚ퟐ = 푭 휞 − ퟐ푴ퟐ 휞 Now put the values of 퐹(Γ) and 푀2(Γ) in 퐼푅퐹(훤) then the result is obtained as = 648푟2 − 1152푟 + 552 − 2(324푟2 − 582푟 + 282) = 12푟 − 12 |풅풆품 (풙 ) − 풅풆품 (풚 )| 푰푹푳푨(횪) = ퟐ ∑ 횪 ퟐ 횪 ퟐ (풅풆품횪(풙ퟐ) + 풅풆품횪(풚ퟐ)) 풙ퟐ풚ퟐ∈푬 |2 − 2| |2 − 3| |3 − 3| = 2 ( ∙ |퐸훼 | + ∙ |퐸훽 | + ∙ |퐸훾 |) 4 {2,2} 5 {2,3} 6 {3,3}

https://biointerfaceresearch.com/ 2975 https://doi.org/10.33263/BRIAC123.29702987 2 = ∙ (12푟 − 12) 5 ퟏ ퟏ ퟐ − − 푰푹푨(횪) = ∑ (풅풆품횪(풙ퟐ) ퟐ − 풅풆품횪(풚ퟐ) ퟐ)

풙ퟐ풚ퟐ∈푬 2 2 2 1 1 훼 1 1 훽 1 1 훾 = ( − ) ∙ |퐸{2,2}| + ( − ) ∙ |퐸 | + ( − ) ∙ |퐸 | √2 √2 √2 √3 {2,3} √3 √3 {3,3} = 0.16832 ∙ (12푟 − 12) ퟏ ퟏ ퟐ 푰푹푩(휞) = ∑ (풅풆품휞(풙ퟐ)ퟐ − 풅풆품휞(풚ퟐ)ퟐ)

풙ퟐ풚ퟐ∈푬 2 훼 2 훽 2 훾 = (√2 − √2) ∙ |퐸{2,2}| + (√2 − √3) ∙ |퐸{2,3}| + (√3 − √3) ∙ |퐸{3,3}| = 0.10106 ∙ (12푟 − 12)

∑풙 풚 ∈푬 √(풅풆품횪(풙ퟐ) ∙ 풅풆품횪(풚ퟐ)) ퟐ풎 푹푹(횪) ퟐ풎 푰푹푪(횪) = ퟐ ퟐ − = − 풎 풏 풎 풏 Now put the values of 푅푅(훤),푛 and 푚 in 퐼푅퐶(훤) then the result is obtained as 108푟2 + (12푟 − 12)√6 − 210푟 + 114 2(27푟2 − 33푟 + 12) = (( ) − ( )) 27푟2 − 33푟 + 12 18푟2 − 18푟 + 6

(12푟3 − 24푟2 + 16푟 − 4)√6 + 27푟4 − 120푟3 + 167푟2 − 96푟 + 22 = ( ) 27푟4 − 60푟3 + 54푟2 − 23푟 + 4 (풅풆품 (풙 ) + 풅풆품 (풚 )) 푰푹푮푨(휞) = ∑ 풍풏 ( 휞 ퟐ 휞 ퟐ ) ퟐ√(풅풆품휞(풙ퟐ) ∙ 풅풆품휞(풚ퟐ)) 풙ퟐ풚ퟐ∈푬 4 훼 5 훽 6 훾 = 푙푛 ( ) ∙ |퐸{2,2}| + 푙푛 ( ) ∙ |퐸 | + 푙푛 ( ) ∙ |퐸 | 4 2√6 {2,3} 6 {3,3} = (0.20391푒 − 1) ∙ (12푛 − 12) ퟏ 푰푹푹 (휞) = ∑ |풅풆품 (풙 ) − 풅풆품 (풚 )| 풕 ퟐ 휞 ퟐ 휞 ퟐ 풙ퟐ풚ퟐ∈푬 1 = ∙ (|2 − 2| ∙ |퐸훼 | + |2 − 3| ∙ |퐸훽 | + |3 − 3| ∙ |퐸훾 |) 2 {2,2} {2,3} {3,3} 1 = ∙ (12푟 − 12) 2

5. Computing the irregularity-based indices for 푫푩푪(풓)

Let 훤 be 퐷퐵퐶(푟). The vertex set and edge set of 퐷퐵퐶(푟) is 푉(훤) = 18푟2 + 6푟 = 푛 and 퐸(훤) = 27푟2 + 3푟 = 푚, where 푚 and 푛 represent the total number of vertices and edges of 퐻퐵퐶(푟) and they are picturized in Figure 2.

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(i) (ii)

(iii) Figure 2. (i) 퐷퐵퐶(2); (ii) 퐷퐵퐶(3); (iii) 퐷퐵퐶(4).

Table 3. Edge partition Table for 퐷퐵퐶(푟).

(푑푒푔훤(푥2) , 푑푒푔훤(푦2)) Total number of edges Set of edges 휆 (2,2) 6푟 | 퐸{2,2}| (2,3) 12푟 휇 | 퐸{2,3}| 2 휈 (3,3) 27푟 − 15푟 | 퐸{3,3}|

Theorem 2: If 훤 is 퐷퐵퐶(푟), then the irregularity-based indices are computed as

6푟−2 1. 푉퐴푅(훤) = ( ) (3푟+1)2

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180푟2−60푟 2. 퐼푅1(훤) = ( ) 3푟+1

(81푟−13)/(9푟+1) (3푟+1−9푟−1) 3. 퐼푅2(훤) = √ 3푟+1 1 4. 퐼푅퐷퐼퐹(훤) = ∙ (12푟) 3 5. 퐴퐿(훤) = 12푟 6. 퐼푅퐿(훤) = 0.40545 ∙ (12푟) 1 7. 퐼푅퐿푈(훤) = ∙ (12푟) 2 1 8. 퐼푅퐿퐹(훤) = ∙ (12푟) √6 9. 퐼푅퐹(훤) = 12푟 2 10. 퐼푅퐿퐴(훤) = ∙ (12푟) 5 11. 퐼푅퐴(훤) = 0.16832 ∙ (12푟) 12. 퐼푅퐵(훤) = 0.10106 ∙ (12푟)

(12푟+4) 6−24푟−12 13. 퐼푅퐶(훤) = ( √ ) 27푟2+12푟+1 14. 퐼푅퐺퐴(훤) = (0.20391푒 − 1) ∙ (12푟) 1 15. 퐼푅푅 (훤) = ∙ (12푟) 푡 2 Proof: Consider 훤 is the 퐷퐵퐶(푟). The total number of vertices and edges of 퐷퐵퐶(푟) are 푉(훤) = 18푟2 + 6푟 and 퐸(훤) = 27푟2 + 3푟. From Table 3, it can be observed that 휆 | 퐸{2,2}| = 6푟

휇 | 퐸{2,3}| = 12푟

휈 2 | 퐸{3,3}| = 27푟 − 15푟.

By using the definitions of 푀1(훤), 푀2(훤), 푅푅(훤) and 퐹(훤), we get the following result as 2 푀1(훤) = 162푟 − 6푟 2 푀2(훤) = 243푟 − 39푟 푅푅(훤) = 81푟2 + 12푟√6 − 33푟 퐹(훤) = 486푟2 − 66푟s ퟐ풎 ퟐ 푴 (휞) ퟐ풎 ퟐ 푽푨푹(휞) = ∑ (풅풆품 (풙 ) − ) = ퟏ − ( ) 휞 ퟐ 풏 풏 풏 풙ퟐ∈풗 Now put the values of 푀1(퐺) , 푛 and 푚 in 푉퐴푅(퐺) then the result is obtained as 2 162푟2 − 6푟 2(27푟2 + 3푟) = ( ) − ( ) 18푟2 + 6푟 18푟2 + 6푟 6푟 − 2 = ( ) (3푟 + 1)2 ퟐ풎 ퟐ풎 푰푹ퟏ(휞) = ∑ 풅풆품 (풙 )ퟑ − ∑ 풅풆품 (풙 )ퟐ = 푭(휞) − 푴 (휞) 휞 ퟐ 풏 휞 ퟐ 풏 ퟏ 풙ퟐ∈풗 풖∈풗 Now put the values of 퐹(퐺), 푀1(퐺) , 푛 and 푚 in 푉퐴푅(퐺) then the result is obtained as https://biointerfaceresearch.com/ 2978 https://doi.org/10.33263/BRIAC123.29702987

2(27푟2+3푟) = (486푟2 − 66푟) − ( ) ∙ (162푟2 − 6푟) 18푟2+6푟 180푟2 − 60푟 = ( ) 3푟 + 1

∑풙 풚 ∈푬 (풅풆품휞(풙ퟐ) ∙ 풅풆품휞(풚ퟐ)) ퟐ풎 푴 (휞) ퟐ풎 푰푹ퟐ(휞) = √ ퟐ ퟐ − = √ ퟐ − 풎 풏 풎 풏

Now put the values of 푀2(훤) , 푛 and 푚 in 퐼푅2(훤) then the result is obtained as 243푟2 − 39푟 2(27푟2 + 3푟) = (√ ) − ( ) 27푟2 + 3푟 18푟2 + 6푟

√(81푟 − 13)/(9푟 + 1) (3푟 + 1 − 9푟 − 1) = 3푟 + 1 ( ) ∑ 풅풆품휞(풙ퟐ) 풅풆품휞(풚ퟐ) 푰푹푫푰푭 휞 = 풙ퟐ풚ퟐ∈푬 | − | 풅풆품휞(풚ퟐ) 풅풆품휞(풚ퟐ) 2 2 2 3 3 3 = | − | ∙ |퐸휆 | + | − | ∙ |퐸휇 | + | − | ∙ |퐸휈 | 2 2 {2,2} 3 3 {2,3} 3 3 {3,3} 1 = ∙ (12푟) 3 ( ) ∑ | ( ) ( )| 푨푳 휞 = 풙ퟐ풚ퟐ∈푬 풅풆품휞 풙ퟐ − 풅풆품휞 풚ퟐ 휆 휇 휈 = |2 − 2| ∙ |퐸{2,2}| + |2 − 3| ∙ |퐸{2,3}| + |3 − 3| ∙ |퐸{3,3}| = 12푟 ( ) ∑ | ( ) ( )| 푰푹푳 휞 = 풙ퟐ풚ퟐ∈푬 풍풏 풅풆품휞 풙ퟐ − 풍풏 풅풆품휞 풚ퟐ 휆 휇 휈 = |푙푛2 − 푙푛2| ∙ |퐸{2,2}| + |푙푛2 − 푙푛3| ∙ |퐸{2,3}| + |푙푛3 − 푙푛3| ∙ |퐸{3,3}| = 0.40545 ∙ (12푟) ( ) ∑ |풅풆품휞(풙ퟐ)−풅풆품휞(풚ퟐ)| 푰푹푳푼 휞 = 풙ퟐ풚ퟐ∈푬 풎풊풏(풅풆품휞(풙ퟐ)∙풅풆품휞(풚ퟐ)) |2−2| |2−3| |3−3| = ∙ |퐸 | + ∙ |퐸휇 | + ∙ |퐸휈 | 2 {2,2} 2 {2,3} 3 {3,3} 1 = ∙ (12푟) 2 ( ) ∑ |풅풆품휞(풙ퟐ)−풅풆품휞(풚ퟐ)| 푰푹푳푭 휞 = 풙ퟐ풚ퟐ∈푬 √(풅풆품휞(풙ퟐ)∙풅풆품휞(풚ퟐ)) |2−2| |2−3| |3−3| = ∙ |퐸휆 | + ∙ |퐸휇 | + ∙ |퐸휈 | √4 {2,2} √6 {2,3} √9 {3,3} 1 = ∙ (12푟) √6 ퟐ 푰푹푭(휞) = ∑ (풅풆품휞(풙ퟐ) − 풅풆품휞(풚ퟐ)) = 푭(휞) − ퟐ푴ퟐ(휞)

풙ퟐ풚ퟐ∈푬 Now put the values of 푀2(훤) and 퐹(훤) in 퐼푅퐹(훤) then the result is obtained as = 486푟2 − 66푟 − 2(243푟2 − 39푟) = 12푟 |풅풆품 (풙 ) − 풅풆품 (풚 )| 푰푹푳푨(휞) = ퟐ ∑ 휞 ퟐ 휞 ퟐ (풅풆품휞(풙ퟐ) + 풅풆품휞(풚ퟐ)) 풙ퟐ풚ퟐ∈푬 |2 − 2| |2 − 3| |3 − 3| = 2 ( ∙ |퐸휆 | + ∙ |퐸휇 | + ∙ |퐸휈 |) 4 {2,2} 5 {2,3} 6 {3,3} 2 = ∙ (12푟) 5

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ퟏ ퟏ ퟐ − − 푰푹푨(휞) = ∑ (풅풆품휞(풙ퟐ) ퟐ − 풅풆품휞(풚ퟐ) ퟐ)

풙ퟐ풚ퟐ∈푬 2 2 2 1 1 휆 1 1 휇 1 1 휈 = ( − ) ∙ |퐸{2,2}| + ( − ) ∙ |퐸 | + ( − ) ∙ |퐸{3,3}| √2 √2 √2 √3 {2,3} √3 √3 = 0.16832 ∙ (12푟) ퟏ ퟏ ퟐ 푰푹푩(휞) = ∑ (풅풆품휞(풙ퟐ)ퟐ − 풅풆품휞(풚ퟐ)ퟐ)

풙ퟐ풚ퟐ∈푬 2 휆 2 휇 2 휈 = (√2 − √2) ∙ |퐸{2,2}| + (√2 − √3) ∙ |퐸{2,3}| + (√3 − √3) ∙ |퐸{3,3}| = 0.10106 ∙ (12푟) ∑ √(풅풆품 (풙 )∙풅풆품 (풚 )) ퟐ풎 푹푹(휞) ퟐ풎 푰푹푪(휞) = 풙ퟐ풚ퟐ∈푬 휞 ퟐ 휞 ퟐ − = − 풎 풏 풎 풏 Now put the values of 푅푅(훤), 푚 and 푛 in 퐼푅퐶(훤) then the result is obtained as 81푟2 + 12푟√6 − 33푟 2(27푟2 + 3푟) = (( ) − ( )) 27푟2 + 3푟 18푟2 + 6푟

(12푟 + 4)√6 − 24푟 − 12 = ( ) 27푟2 + 12푟 + 1 (풅풆품 (풙 ) + 풅풆품 (풚 )) 푰푹푮푨(휞) = ∑ 풍풏 ( 휞 ퟐ 휞 ퟐ ) ퟐ√(풅풆품휞(풙ퟐ) ∙ 풅풆품휞(풚ퟐ)) 풙ퟐ풚ퟐ∈푬 4 5 휇 6 휈 = 푙푛 ( ) ∙ |퐸{2,2}| + 푙푛 ( ) ∙ |퐸 | + 푙푛 ( ) ∙ |퐸{3,3}| 4 2√6 {2,3} 6 = (0.20391푒 − 1) ∙ (12푟) ퟏ 푰푹푹 (휞) = ∑ |풅풆품 (풙 ) − 풅풆품 (풚 )| 풕 ퟐ 휞 ퟐ 휞 ퟐ 풙ퟐ풚ퟐ∈푬 1 = ∙ (|2 − 2| ∙ |퐸휆 | + |2 − 3| ∙ |퐸휇 | + |3 − 3| ∙ |퐸휈 |) 2 {2,2} {2,3} {3,3} 1 = ∙ (12푟) 2

6. Computing irregularity-based indices for 풄푯푩푪(풓)

Let 훤 be 푐퐻퐵퐶(푟). The vertex set and edge set of 푐퐻퐵퐶(푟) is 푉(훤) = 24푟2 − 30푟 + 12 = 푛 and 퐸(훤) = 36푟2 − 54푟 + 24 = 푚 , where 푚 and 푛 represent the total number of vertices and edges of 퐻퐵퐶(푟) and they are picturized in Figure 3.

(i) (ii)

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(iii) (iv)

Figure 3. (i) Coronene; (ii) 푐퐻퐵퐶(2); (iii) 푐퐻퐵퐶(3); (iv) 푐퐻퐵퐶(4).

Table 4. Edge partition Table for 푐퐻퐵퐶(푟).

(푑푒푔훤(푥2) , 푑푒푔훤(푦2)) Total number of edges Set of edges (2,2) 12푟 − 6 𝜍 |퐸{2,2}| 휏 (2,3) 12푟 − 12 |퐸{2,3}| 2 휐 (3,3) 36푟 − 78푟 + 42 |퐸{3,3}|

Theorem 3. If Γ is the 푐퐻퐵퐶(푟) network, then the irregularity-based indices are computed as

2 1. 푉퐴푅(훤) = (24푟 −42푟+20) 4푟2−5푟+2 120(푟−1)2(3푟−2) 2. 퐼푅1(훤) = ( ) 4푟2−5푟+2

√54푟2−97푟+47⁄6푟2−9푟+4∙(4푟2−5푟+2)−12푟2+18푟−8 3. 퐼푅2(훤) = ( ) 4푟2−5푟+2 1 4. 퐼푅퐷퐼퐹(훤) = ∙ (12푟 − 12) 3 5. 퐴퐿(훤) = 12푟 − 12 6. 퐼푅퐿(훤) = 0.40545 ∙ (12푟 − 12) 1 7. 퐼푅퐿푈(훤) = ∙ (12푟 − 12) 2 1 8. 퐼푅퐿퐹(훤) = ∙ (12푟 − 12) √6 9. 퐼푅퐹(훤) = 12푟 − 12 2 10. 퐼푅퐿퐴(훤) = ∙ (12푟 − 12) 5 11. 퐼푅퐴(훤) = 0.16832 ∙ (12푟 − 12) 12. 퐼푅퐵(훤) = 0.10106 ∙ (12푟 − 12)

(푟−1)(8 6푟2−10 6푟−14푟2+4 6+15푟−6) 13. 퐼푅퐶(훤) = ( √ √ √ ) 24푟4−66푟3+73푟2−38푟+8 14. 퐼푅퐺퐴(훤) = (0.20391푒 − 1) ∙ (12푟 − 12)

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1 15. 퐼푅푅 (훤) = ∙ (12푟 − 12) 푡 2 Proof: Consider 훤 is the 푐퐻퐵퐶(푟).The total number of vertices and edges of 푐퐻퐵퐶(푟) are 푉(훤) = 24푟2 − 30푟 + 12 and 퐸(훤) = 36푟2 − 54푟 + 24. From Table 4, it can be observed that

𝜍 |퐸{2,2}| = 12푟 − 6

휏 |퐸{2,3}| = 12푟 − 12 휐 2 |퐸{3,3}| = 36푟 − 78푟 + 42.

By using the definitions of 푀1(훤), 푀2(훤), 푅푅(훤) and 퐹(훤), we get the following result as 2 푀1(훤) = 216푟 − 360푟 + 168 2 푀2(훤) = 324푟 − 582푟 + 282 푅푅(훤) = 108푟2 + (12푟 − 12)√6 − 210푟 + 114 퐹(훤) = 648푟2 − 1152푟 + 552 ퟐ풎 ퟐ 푴 (푮) ퟐ풎 ퟐ 푽푨푹(휞) = ∑ (풅풆품 (풙 ) − ) = ퟏ − ( ) 푮 ퟐ 풏 풏 풏 풖∈풗 Now put the values of 푀1(훤) , 푛 and 푚 in 푉퐴푅(훤) then the result is obtained as 2 216푟2 − 360푟 + 168 2(36푟2 − 54푟 + 24) = ( ) − ( ) 24푟2 − 30푟 + 12 24푟2 − 30푟 + 12 24푟2 − 42푟 + 20 = ( ) 4푟2 − 5푟 + 2 ퟐ풎 ퟐ풎 푰푹ퟏ(휞) = ∑ 풅풆품 (풙 )ퟑ − ∑ 풅풆품 (풙 )ퟐ = 푭(휞) − 푴 (휞) 휞 ퟐ 풏 휞 ퟐ 풏 ퟏ 풙ퟐ∈풗 풙ퟐ∈풗 Now put the values of 퐹(훤), 푀1(훤) , 푛 and 푚 in 퐼푅1(훤) then the result is obtained as 2(36푟2 − 54푟 + 24) = 648푟2 − 1152푟 + 552 − ( ) (216푟2 − 360푟 + 168) 24푟2 − 30푟 + 12 120(푟 − 1)2(3푟 − 2) = ( ) 4푟2 − 5푟 + 2

∑풙 풚 ∈푬 (풅풆품휞(풙ퟐ) ∙ 풅풆품휞(풚ퟐ)) ퟐ풎 푴 (휞) ퟐ풎 푰푹ퟐ(횪) = √ ퟐ ퟐ − = √ ퟐ − 풎 풏 풎 풏

Now put the values of 푀2(훤) , 푛 and 푚 in 퐼푅2(훤) then the result is obtained as 324푟2 − 582푟 + 282 2(36푟2 − 54푟 + 24) = (√ ) − ( ) 36푟2 − 54푟 + 24 24푟2 − 30푟 + 12

√54푟2 − 97푟 + 47⁄6푟2 − 9푟 + 4 ∙ (4푟2 − 5푟 + 2) − 12푟2 + 18푟 − 8 = ( ) 4푟2 − 5푟 + 2 ( ) ∑ 풅풆품휞(풙ퟐ) 풅풆품휞(풚ퟐ) 푰푹푫푰푭 휞 = 풙ퟐ풚ퟐ∈푬 | − | 풅풆품휞(풚ퟐ) 풅풆품휞(풚ퟐ) 2 2 2 3 3 3 = | − | ∙ |퐸 | + | − | ∙ |퐸 | + | − | ∙ |퐸 | 2 2 {2,2} 3 3 {2,3} 3 3 {3,3} 1 = ∙ (12푟 − 12) 3

https://biointerfaceresearch.com/ 2982 https://doi.org/10.33263/BRIAC123.29702987 ( ) ∑ | ( ) ( )| 푨푳 휞 = 풙ퟐ풚ퟐ∈푬 풅풆품휞 풙ퟐ − 풅풆품휞 풚ퟐ = |2 − 2| ∙ |퐸{2,2}| + |2 − 3| ∙ |퐸{2,3}| + |3 − 3| ∙ |퐸{3,3}| = 12푟 − 12 ( ) ∑ | ( ) ( )| 푰푹푳 휞 = 풙ퟐ풚ퟐ∈푬 풍풏 풅풆품휞 풙ퟐ − 풍풏 풅풆품휞 풚ퟐ = |푙푛2 − 푙푛2| ∙ |퐸{2,2}| + |푙푛2 − 푙푛3| ∙ |퐸{2,3}| + |푙푛3 − 푙푛3| ∙ |퐸{3,3}| = 0.40545 ∙ (12푟 − 12) ( ) ∑ |풅풆품휞(풙ퟐ)−풅풆품휞(풚ퟐ)| 푰푹푳푼 휞 = 풙ퟐ풚ퟐ∈푬 풎풊풏(풅풆품휞(풙ퟐ)∙풅풆품휞(풚ퟐ)) |2−2| |2−3| |3−3| = ∙ |퐸 | + ∙ |퐸 | + ∙ |퐸 | 2 {2,2} 2 {2,3} 3 {3,3} 1 = ∙ (12푟 − 12) 2 ( ) ∑ |풅풆품휞(풙ퟐ)−풅풆품휞(풚ퟐ)| 푰푹푳푭 휞 = 풙ퟐ풚ퟐ∈푬 √(풅풆품휞(풙ퟐ)∙풅풆품휞(풚ퟐ)) |2−2| |2−3| |3−3| = ∙ |퐸{ }| + ∙ |퐸{ }| + ∙ |퐸{ }| √4 2,2 √6 2,3 √9 3,3 1 = ∙ (12푟 − 12) √6 ( ) ∑ ( ( ) ( ))ퟐ ( ) ( ) 푰푹푭 휞 = 풙ퟐ풚ퟐ∈푬 풅풆품휞 풙ퟐ − 풅풆품휞 풚ퟐ = 푭 휞 − ퟐ푴ퟐ 휞 Now put the values of 퐹(훤) , 푀2(훤) in 퐼푅퐹(훤) then the result is obtained as = 648푟2 − 1152푟 + 552 − 2(324푟2 − 582푟 + 282) = 12푟 − 12 |풅풆품 (풙 ) − 풅풆품 (풚 )| 푰푹푳푨(휞) = ퟐ ∑ 휞 ퟐ 휞 ퟐ (풅풆품휞(풙ퟐ) + 풅풆품휞(풚ퟐ)) 풙ퟐ풚ퟐ∈푬 |2 − 2| |2 − 3| |3 − 3| = 2 ( ∙ |퐸 | + ∙ |퐸 | + ∙ |퐸 |) 4 {2,2} 5 {2,3} 6 {3,3} 2 = ∙ (12푟 − 12) 5 ퟏ ퟏ ퟐ − − 푰푹푨(휞) = ∑ (풅풆품휞(풙ퟐ) ퟐ − 풅풆품휞(풚ퟐ) ퟐ)

풙ퟐ풚ퟐ∈푬 1 1 2 1 1 2 1 1 2 = ( − ) ∙ |퐸{2,2}| + ( − ) ∙ |퐸{2,3}| + ( − ) ∙ |퐸{3,3}| √2 √2 √2 √3 √3 √3 = 0.16832 ∙ (12푟 − 12) ퟏ ퟏ ퟐ 푰푹푩(휞) = ∑ (풅풆품휞(풙ퟐ)ퟐ − 풅풆품휞(풚ퟐ)ퟐ)

풙ퟐ풚ퟐ∈푬 2 2 2 = (√2 − √2) ∙ |퐸{2,2}| + (√2 − √3) ∙ |퐸{2,3}| + (√3 − √3) ∙ |퐸{3,3}| = 0.10106 ∙ (12푟 − 12)

∑풙 풚 ∈푬 √(풅풆품휞(풙ퟐ) ∙ 풅풆품휞(풚ퟐ)) ퟐ풎 푹푹(휞) ퟐ풎 푰푹푪(휞) = ퟐ ퟐ − = − 풎 풏 풎 풏 Now put the values of 푅푅(훤) , 푛 and 푚 in 퐼푅퐶(훤) then the result is obtained as 108푟2 + (12푟 − 12)√6 − 210푟 + 114 2(36푟2 − 54푟 + 24) = (( ) − ( )) 36푟2 − 54푟 + 24 24푟2 − 30푟 + 12

(푟 − 1)(8√6푟2 − 10√6푟 − 14푟2 + 4√6 + 15푟 − 6) = ( ) 24푟4 − 66푟3 + 73푟2 − 38푟 + 8

https://biointerfaceresearch.com/ 2983 https://doi.org/10.33263/BRIAC123.29702987 (풅풆품 (풙 ) + 풅풆품 (풚 )) 푰푹푮푨(휞) = ∑ 풍풏 ( 휞 ퟐ 휞 ퟐ ) ퟐ√(풅풆품휞(풙ퟐ) ∙ 풅풆품휞(풚ퟐ)) 풙ퟐ풚ퟐ∈푬 4 5 6 = 푙푛 ( ) ∙ |퐸{2,2}| + 푙푛 ( ) ∙ |퐸{2,3}| + 푙푛 ( ) ∙ |퐸{3,3}| 4 2√6 6 = (0.20391푒 − 1) ∙ (12푟 − 12) ퟏ 푰푹푹 (휞) = ∑ |풅풆품 (풙 ) − 풅풆품 (풚 )| 풕 ퟐ 휞 ퟐ 휞 ퟐ 풙ퟐ풚ퟐ∈푬 1 = ∙ (|2 − 2| ∙ |퐸 | + |2 − 3| ∙ |퐸 | + |3 − 3| ∙ |퐸 |) 2 {2,2} {2,3} {3,3} 1 = ∙ (12푟 − 12) 2

7. Graphical Representations of the Obtained Results

The chemical structures are visualized as molecular graphs by employing the graph- theoretical methods. Based on the molecular graphs, the irregularity descriptors are computed as functions of 푟, the parameter which defines the underlying molecular topology. To understand the relationship and behavioral pattern of the calculated indices, the analytical expressions of a few selected indices are represented as 3D plots against variable 푟. The 3D plots of Theorems 1, 2, and 3 are provided in Figures 4, 5, and 6, respectively. It is obvious from the graphs that the indices vary depending upon the molecular structure.

Figure 4. 3D Plots for the results obtained in Theorem 1

https://biointerfaceresearch.com/ 2984 https://doi.org/10.33263/BRIAC123.29702987

Figure 5. 3D Plots for the results obtained in Theorem 2.

Figure 6. 3D Plots for the results obtained in Theorem 3.

8. Conclusions

In this article, the analytical expressions of irregularity indices have been computed for three types of PAH structures, namely, hexaperi-hexabenzocoronene (HBC), dodeca-benzo- circumcoronene (DBC), and hexa-cata-hexabenzocoronene (cHBC). The 3D graphical https://biointerfaceresearch.com/ 2985 https://doi.org/10.33263/BRIAC123.29702987 representations of the analytical expressions provide a visual conceptualization of the relationship between the irregularity indices and the corresponding molecular structures. The computation of eccentricity-based indices for these structures can also contribute to further studies on these structures, as they have not been explored before. This remains an open problem for future researchers.

Funding

This research received no external funding.

Acknowledgment

The authors wish to thank the management of Sri Sivasubramaniya Nadar College of Engineering, Kalavakkam-603110, for their continuous support and encouragement to carry out this research work.

Conflict of interest

The authors declare that there is no conflict of interest regarding the publication of this article.

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