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.