Int. J. Anal. Appl. (2023), 21:66 Topological Evaluation of Four Para-Line Graphs Absolute Pentacene Graphs Using Topological Indices Mukhtar Ahmad1, Muhammad Jafar Hussain2, Gulnaz Atta3, Sajid Raza2, Irfan Waheed2, Ather Qayyum2,∗ 1Department of Mathematics, Khawaja Fareed University of Engineering and Information Technology Rahim Yar Khan, Pakistan 2Department of Mathematics, Institute of Southern Punjab Multan, Pakistan 3Department of Mathematics, University of Education Lahore DGK Compuse, Pakistan ∗Corresponding author: atherqayyum@isp.edu.pk Abstract. A real-number to molecular structure mapping is a topological index. It is a graph invariant method for describing physico-chemical properties of molecular structures specific substances. In that article, We examined pentacene’s chemical composition. The research on the subsequent indices is reflected in our paper, we conducted an analysis of several indices including general randic connectivity index, first general zagreb index, general sum-connectivity index, atomic bond connectivity index, geometric-arithmetic index, fifth class of geometric-arithmetic indices, hyper-zagreb index, first and second multiple zagreb indices for a four para-lines graphs of linear [n]-pentacene and multi-pentacene. 1. Introduction and preliminaries All substances molecule possesses qualities, both chemical and physical, and certain may also exhibit physiologically active characteristics. Several pharmaceutical companies are really hunting for novel antibacterial chemicals. For this reason, hundreds of compounds are examined, however costly exam- inations for biology. In order to circumvent such issue, additional methods for investigating potential antibiotics employ the relationship between structural features and biological activity or features of chemical and physical nature. Topological indices, or molecular descriptors, provide insights into the physicochemical properties of molecules. They are valuable tools for understanding and explaining Received: May 22, 2023. 2020 Mathematics Subject Classification. 92C40. Key words and phrases. topological indices; graphs of four para-lines; nanostructures; pentacene. https://doi.org/10.28924/2291-8639-21-2023-66 ISSN: 2291-8639 © 2023 the author(s). https://doi.org/10.28924/2291-8639-21-2023-66 2 Int. J. Anal. Appl. (2023), 21:66 the characteristics of chemical compounds. Several graph invariants have been created in recent years and have been used in many academic fields such as structural chemistry, theoretical chemistry, environmental chemistry, toxicology, and pharmacology. Because of the substantial industrial need, re- searchers are urged to study topological indices. More than 400 topological indexes have been opened a consequence of research. Chemical compounds’ topological structures and chemical characteristics are tightly related, since each compound’s shape is critical to determining its functionality. Topolog- ical indices are often used in multilinear regression modelling, chemical documentation, drug design, QSAR/QSPR modelling, and database selection. Molecular descriptors are utilized to describe the physicochemical properties of molecular structures. These descriptors can be classified into three main types. degree-based indices [1–5], distance-based indices [6–11] and spectrum-based indices [12–15]. Studies that have been documented in the literature (see [16–18]) use indicators that are based on both distances and degrees. Due to pentacene’s important functions in both electrical devices and organic solar cells, a popular hydrocarbon semiconductor, it is necessary to optimise organic solar cells for less expensive energy sources [19]. The Georgia Institute of Technology researchers have developed method to produce portable artificial solar cells. Pentacene has been shown to be a very efficient means of converting sunlight into energy. In contrast to other materials, pentacene functions well as a semiconductor due to its crystalline properties. Pentacene’s relevance motivated us to do topological study on it, and as a result, we have made several important discoveries that could be helpful for analysing pentacene’s physical and chemical characteristics. See [20,21] for further topological research on pentacene. Consider an easy graph G consisting of a edge set E(G) and vertex set V (G), where loops and several edges present are excluded. The set x ∈ V (G),Nx of neighbors in G is represented by Nx, and the valence (degree) of x is equal to dx1 = |Nx| and Sx1 = ∑ y∈Nx dx2. By inserting a vertex between every edge of the given graph, the edges are divided into two, resulting in the graph being subdivided. This operation, known as graph subdivision and denoted as S(G), leads to the formation of a line graph where adjacent edges in G become connected vertices in the new graph. The resulting graph, denoted as L(G), represents the line graph of the subdivision graph. In this article, the four para-line graph of G is represented by L(S(G)) (referred to as G?). Conversely, G? can be constructed from G using the following procedure: 1. Replace each vertex x1 ∈ V (G) with Kx1, complete graph on dx1 vertices; 2. There is an edge connecting the vertex Kx1 and the vertex Kx2 in G ? if and only if there is an edge that coincides with x1 and x2 in G; 3. For each vertex x2 in Kx1, in G ?, the valency (degree) of x2 is equal to the valency (degree) of x1 in G. Structural chemistry commonly utilizes these diagrams. The research focus on four para-line graphs has diminished in recent times, but there is a shift happening. One appealing aspect of these graphs Int. J. Anal. Appl. (2023), 21:66 3 is their straightforward construction process. The carbon skeleton, in which each atom acts as the vertex and each link between nearby atoms as the edge, may be used to generate any chemical compound. For example, Butane is an organic compound with the formula (C4H10). Butane is a saturated hydrocarbon containing 4 of carbon atoms, with an unbranched structure. Butane is mainly used as a gasoline blend, alone or mixed with propane. It is also used as a feedstock for the production of ethylene and butadiene. Butane, like propane, is obtained from natural gas or refineries, and the two gases usually occur together. Butane is stored under pressure as a liquid. When the curler is turned on, butane is released and turns into a gas. Figure 1(a) depicts the the molecular graph and its structure of butane. Furthermore, Figure 2(b) and (c) exhibit the four para-line graphs derived from the molecular plot of butane. now figure is; Figure 1. (a) The molecular architecture of butane Figure 2. (b) The molecular architecture of butane (c) four para-line graph of butane to accurately represent it The general randic connectivity index G is defined as [12]. Rα(G) = ∑ x1x2∈E(G) (dx1dx2) α (1.1) The first universal Zagreb index was presented by Li and Zhao [22]: Mα(G) = ∑ x1∈V (G) (dx1) α (1.2) 4 Int. J. Anal. Appl. (2023), 21:66 The the general sum connectivity index of the G chart was introduced in 2010 [23]: χα(G) = ∑ x1x2∈E(G) (dx1 + dx2) α (1.3) The index (ABC) was proposed by Estrada [24]. It is expressed as follows for a graph G: ABC(G) = ∑ x1x2∈E(G) √ dx1 + dx2 − 2 dx1dx2 (1.4) The geometric-arithmetic index (GA) was introduced by Vukicevic and Furtula [25]. It is denoted as GA and is defined as follows for a graph G resently A. Asghar et.al [31]: GA(G) = ∑ x1x2∈E(G) 2 √ (dx1dx2) (dx1 + dx2) (1.5) Ghorbani et al. [26] described another index belonging to the 4th class of indices, denoted as (ABC), which is defined as follows resently Zaib Hassan Niazi et.al [32]: ABC4(G) = ∑ x1x2∈E(G) √ dS1 + Sx2 − 2 Sx1Sx2 (1.6) Graovac et al. [27] introduced a fifth class of geometric-arithmetic indices denoted as GA5, which is defined as follows: GA5(G) = ∑ x1x2∈E(G) 2 √ (Sx1Sx2) (Sx1 + Sx2) (1.7) Established the hyper-zagreb index in 2013 as follows resently Mukhtar Ahmad et.al [33]: HM(G) = ∑ x1x2∈E(G) (dx1 + dx2) 2 (1.8) In 2012, Ghorbani and Azimi introduced two new types of zagreb graph indices. The first is the first multiple zagreb index, denoted as PM1(G). The second multiple zagreb index is used, denoted as PM2(G). Additionally, the first and second zagreb polynomials, M1(G,p) and M2(G,p), respectively, are characterised as: PM1(G) = Πx1x2∈E(G)(dx1 + dx2) (1.9) PM2(G) = Πx1x2∈E(G)(dx1 ×dx2) (1.10) M1(G,p) = ∑ x1x2∈E(G) P(dx1+dx2) (1.11) M2(G,p) = ∑ x1x2∈E(G) P(dx1×dx2) (1.12) Int. J. Anal. Appl. (2023), 21:66 5 2. Topological index of four para-line graphs For an index that Schultz offered, Ranjini created the independent relations. Under the watchful eye of the Schultz index, these researchers looked at the subdivision of a number of graphs, including helm, ladder, tadpole, and wheel [28]. They also looked at the ladder, tadpole, and wheel four para-line graph under the zagreb index [29]. In 2015, Xu and Su conducted an analysis of two indices specific to ladder, tadpole, and wheel graphs constructed using tare lines and named the total connectivity index of the sum and the co-index [30]. Nadim et al. calculated the atomic bond connectivity index and fifth class of geometric arithmetic indices for four para-line tadpole, wheel, and ladder graphs. They also investigated several other indices, including randic general connectivity index, first zagreb general index, summation general connectivity index, atomic bond connectivity index, geometric arithmetic index, fifth class of geometric arithmetic indices, hyperzagreb index, the first and second multiple zagreb index for a four paralinear graphs of linear [n]-pentacene and multiple pentacene., lattice plot in nanotorus TUC4C8[p,q] and 2D nanotube. In our study, we computed various indices, including randic general connectivity index, first za- greb general index, summation general connectivity index, atomic bond connectivity index, geometric arithmetic index, fifth class of geometric arithmetic indices, hyperzagreb index. 2.1. Molecular characteristics of the linear [n]-pentacene four para-line graph. Figure 3 depicts the linear [n]-pentacene molecular graph, which is indicated by the symbol Tn. Tn consists of 28n− 2 edges and 22n vertices. Theorem 2.1. Consider a four para-line graph G? derived from the graph Tn. Mα(G ?) = (5n + 2)2α+2 + 3α+1(12n− 4). Proof. In Figure 3, the graph G? is displayed. There are 56n − 4 vertices in total in G?, this has 36n− 12 vertices of degree and 20n + 8 vertices of degree, where Mα(G ?) = (5n + 2)2α+2 + 3α+1(12n− 4). Theorem 2.2 Consider a four para-line graph G? derived from the graph Tn. 1. Rα(G?) = (10n + 10)16α + (20n− 4)20α + (44n− 16)25α. 2. χα(G?) = (10n + 10)8α + (20n− 4)9α + (44n− 16)10α. 3. ABC(G?) = (15 √ 2 + 88 3 )n + 3 √ 2 − 32 3 . 4. GA(G?) = (54 + 8 √ 6)n− 6 − 8 5 √ 6. Proof. The total number of edges in G? is determined by the formula 74n− 10. The edges in G? can be divided into three sets, E1(G?), E2(G?), and E3(G?), which do not intersect with each other. The edge partition E1(G?) contains 10n + 10 edges x1,x2, where dx1 = dx2 = 4, edge the partition E2(G ?) contains 20n − 4 edges x1,x2, where dx1 = 4 and dx2 = 5, and The edge partition E3(G ?) consists of 44n−16 edges. This partition includes edges x1 and x2, where dx1 = dx2 = 5. By utilizing we get the required outcomes using formulae (1), (3), (4), and (5). Theorem 2.3 Consider a four para-line graph G? derived from the graph Tn. 6 Int. J. Anal. Appl. (2023), 21:66 Figure 3. Linear Pentacene 1. ABC 4 (G?) = ( √ 110 + 4 √ 2 + 2 √ 30 + 16 3 )n + 5 2 + 2 5 − 8 5 √ 2 − 2 3 √ 30 − 1 5 √ 110 − 32 9 2. GA5(G?) = (30 + 8013 √ 10 + 288 17 √ 2)n− 2 + 16 9 √ 5 − 16 13 √ 10 − 96 17 √ 2 Proof. Assuming that the set of edges depends on the sum of the degrees of the neighbors of the end vertices, we can partition edges that divide (G?) into seven distinct sets: E6(G?), E7(G?), ..., E12(G ?). Thus, we have E(G?) = ⋃12 i=6Ei (G ?). The edge assortment E6(G?) comprises 12 edges x1x2, where Sx1 = Sx2 = 6, the edge collection E7(G ?) holds 6 edges x1x2, where Sx1 = 6 and Sx2 = 7, the edge collection E8(G?) holds 11n − 5 edges x1x2, where Sx1 = Sx2 = 7, set of edges E9(G ?) contains 22n− 5 edges x1x2, where Sx1 = 7 and Sx2 = 10, edge the collection E10(G ?) contains 10n edges x1x2, where Sx1 = Sx2 = 10, the edge set E11(G ?) contains 26n − 9 edges x1x2, where Sx1 = 10 and Sx2 = 11 and the set of edges E12(G ?) is satisfied 13n− 9 edges x1x2, where Sx1 = Sx2 = 11. By utilizing we can get the required outcomes using formulae 6 and 7. Theorem 2.4 Consider a four para-line graph G? derived from the graph Tn 1. HM(G) = 6480n− 1464. 2. PM1(G?) = 810n+10 × 920n−4 × 1044n−16. 3. PM2(G) = 1610n+10 × 2020n−4 × 2544n−16. Proof. Consider a four para-line graph G? of a linear pentacene. Based on the angles of the final vertex, the collection of edges E(G?) might be categorised as three distinct groups. The first category, E1(G ?), consists of 10n + 10 edges x1x2, where dx1 = dx2 =4. The second category, E2(G ?), includes 20n−4 edges x1x2, where dx1 = 4 and dx2 =5. The third category, E3(G ?), comprises 44n−16 edges x1x2, where dx1 = dx2 = 5. Let |E1(G)| = e4,4, |E2(G)| = e4,5, and |E3(G)| = e5,5. Therefore, 1. HM(G) = ∑ x1x2∈E(G)(dx1 + dx2) 2 HM(G) = ∑ x1x2∈E1(G)[dx1 + dx2] 2 + ∑ x1x2∈E2(G)[dx1 + dx2] 2+ ∑ x1x2∈E3(G)[dx1 + dx2] 2 HM(G) = 64|E1(G)| + 81|E2(G)| + 100|E3(G)| HM(G) = 64(10n + 10) + 81(20n− 4) + 100(44n− 16) HM(G) = 460n + 460 + 1620n− 324 + 4400n− 1600 This implies that HM(G) = 6480n− 1464. 2. PM1(G) = Πx1x2∈E1(G)(dx1 + dx2) × Πx1x2∈E2(G)(dx1 + dx2) × Πx1x2∈E3(G)(dx1 + dx2) PM1(G) = 8|E1(G)| × 9|E2(G)| × 10|E1(G)| PM1(G) = 810n+10 × 920n−4 × 1044n−16 Int. J. Anal. Appl. (2023), 21:66 7 3. PM2(G) = Πx1x2∈E1(G)(dx1 ×dx2) × Πx1x2∈E2(G)(dx1 × (dx2) × Πx1x2∈E3(G)(dx1 ×dx2) PM2(G) = 16|E1(G)| × 20|E2(G)| × 25|E1(G)| PM2(G) = 16|E1(G)| × 20|E2(G)| × 25|E1(G)| PM2(G) = 1610n+10 × 2020n−4 × 2544n−16 2.2. Molecular descriptors of four paraline graphs for multiple pentacenes. The chemical diagram Tm,n representing multiple pentacene is depicted in Figure 4. This graph consists of 22mn vertices and 33mn− 2m− 5n edges. Theorem 2.5 Consider a four para-line graph G? derived from the graph Tm,n. Mα(G?) = (5n + 2)2α+2 + 3α+1(12n− 4). Proof. Figure 5 shows the graph G? in a visual format. It has 56n−4 worth of vertices in total, of which 20n + 8 and 36n− 12 have degrees of 3 and 4, respectively. Using formula 2, we can calculate Mα(G?). Theorem 2.6 Consider a four para-line graph G? derived from the graph Tm,n. 1. Rα(G?) = (10n + 6m + 4)16α+ (4m + 20n− 8)20α + (99mn− 20m− 55n + 4)25α. 2. χα(G?) = (10n + 6m + 4)8α + (4m + 20n− 8)9α + (99mn− 20m− 55n + 4)10α. 3. ABC(G?) = (15 √ 2 - 110 3 )n + (5 √ 2 - 40 3 )m− 2 √ 2 + 66mn + 8 3 . 4. GA(G?) = (−45 + 8 √ 6)n+(8 5 √ 6 − 14)m + 99mn+ 8 - 16 5 √ 6. Proof. The division graph S(Tm,n) comprises a total of 198mn − 20m − 50 vertices and 99mn − 10m − 25n edges. There are 8m + 20nverticesof degree2and66mn-12m-30n vertices of degree 3, according to the vertex division. The edge set E(G?) of the four para-line graph G? consists of 99mn − 20m − 55n + 4 edges. Based on the angles of the end vertices, these edges are divided into three groups, i.e, E(G?) = E1(G?) ∪E2(G?) ∪E3(G?). The edge separation E1(G?) consists of 10n+ 6m+ 4 edges x1x2. where dx1 = dx2 = 4. Edge Separation 4m+ 20n−8 with E2(G ?) Edge x1x2, where dx1 = 4 and dx2 = 5. Lastly, Separating the edges E3(G ?) comprises 99mn − 20m − 55n + 4 edges x1x2, where dx1 = dx2 = 5. By applying the required outcome may be produced using formulae (1), (3), (4) and (5). Figure 4. Multiple Pentacene Theorem 2.7 Consider a four para-line graph G? derived from the graph Tm,n. 1. ABC4(G?) = (44m + √ 14 + 4 √ 2 + √ 110 + 2 √ 30 − 116 3 )n + (1 2 √ 6 8 Int. J. Anal. Appl. (2023), 21:66 +1 5 √ 110 + 2 5 √ 35 − 112 9 + 2 3 √ 30)m + 2 √ 6 − 8 5 √ 2 − 2 5 √ 110 − 4 3 √ 30 + 80 9 . 2. GA5(G?) = (8013 √ 10 + 99m + 288 17 √ 2 − 69)n + (−26 + 16 13 √ 10 + 16 9 √ 5 + 96 17 √ 2)m− 192 17 √ 10 − 32 13 √ 10 + 24 Proof. Seven distinct edge sets may be formed from the set of edges by taking into account the degree sum of end vertices’ neighbours. Ei (G?), where i = 6, 7, ..., 12. Thus, we have E(G?) =⋃12 i=6Ei (G ?). The edge partition E6(G?) contains 2m + 8 edges x1x2, where Sx1 = Sx2 = 6. The edge partition E7(G?) consists of 4m edges x1x2, where Sx1 = 6 and Sx2 = 7. Edge partition E8(G ?) contains 10n − 4 edges x1x2. where Sx1 = Sx2 = 7. Edge partition E9(G ?) contains 20n + 4m − 8 edges x1x2. where Sx1 = 8 and Sx2 = 9. Edge partition E10(G ?) consists of 10n edges x1x2. where Sx1 = Sx2 = 9. Edge partition E11(G ?) contains 8m + 24n − 16 edge x1x2. where Sx1 = 10 and Sx2 = 11. Finally, edge partition E12(G ?) contains 99mn − 28m − 87n + 20 edge x1x2. where Sx1 = Sx2 = 11. By utilizing formulas (6) and (7), we obtain the desired result. Figure 5. Four para-line graph multiple of pentacene By performing computations on the chemical structures of multiple-pentacene, we obtain the fol- lowing indices: HM(G),PM1(G),PM2(G). Theorem 2.8 Consider a four para-line graph G? derived from the graph Tm,n. 1. HM(G?) = 9900mn− 1292m− 3420n + 8 2. PM1(G?) = 810n+6m+4 × 94m+20n−8 × 1099mn−20m−55n+4. 3. PM2(G?) = 1610n+6m+4 × 204m+20n−8 × 2599mn−20m−55n+4 4. M1(G,p) = (10n + 6m + 4)P8 + (4m + 20n− 8)P9 + (99mn− 20m− 55n + 4)P10. 5. M2(G,p) = (10n + 6m + 4)P16 + (4m + 20n− 8)P20 + (99mn− 20m− 55n + 4)P25. Proof. Consider a graph G? with its edges broken down into three parts categories due to the degrees of the final vertex. The initial category, denoted as E1(G), consists of 10n + 6m + 4 edges x1x2, which both vertices x1 and x2 have a degree of 4. The second category, denoted as E2(G), contains 4m + 20n − 8 edges x1x2, which x1 has a degree of 4 and x2 has a degree of 5. The third category, denoted as E3(G), includes 99mn− 20m− 55n + 4 edges x1x2, where both vertices x1 and x2 have a degree of 5. We can observe that the cardinality of E1(G) is equal to e4,4, E2(G) is equal to e4,5, and E3(G) is equal to e5,5. 1. HM(G?) = ∑ x1x2∈E(G)(dx1 + dx2) 2 HM(G?) = ∑ x1x2∈E1(G)[dx1 + dx2] 2 + ∑ x1x2∈E2(G)[dx1 + dx2] 2 + ∑ x1x2∈E3(G)[dx1 + dx2] 2 Int. J. Anal. Appl. (2023), 21:66 9 HM(G?) = 64|E1(G)| + 81|E2(G)| + 100|E3(G)| HM(G?) = 64(10n + 6m + 4) + 81(4m + 20n− 8) + 100(99mn− 20m− 55n + 4) HM(G?) = 460n + 384m + 256 + 324m + 1620n− 648 +9900mn− 2000m− 5500n + 400 This implies that HM(G?) = 9900mn− 1292m− 3420n + 8 Since, 2. PM1(G?) = Πx1x2∈E(G)(dx1 + dx2) PM1(G ?) = Πx1x2∈E1(G)(dx1 + dx2) × Πx1x2∈E2(G)(dx1 + dx2) × Πx1x2∈E3(G)(dx1 + dx2) PM1(G ?) = 810n+6m+4 × 94m+20n−8 × 1099mn−20m−55n+4. Now that 3. PM2(G?) = Πx1x2∈E(G)(dx1 ×dx2) PM2(G ?) = Πx1x2∈E1(G)(dx1 timesdx2) × Πx1x2∈E2(G)(dx1 ×dx2) × Πx1x2∈E3(G)(dx1 ×dx2) PM2(G ?) = 16|E1(G)| × 20|E1(G)| × 25|E1(G)| PM2(G ?) = 1610n+6m+4 × 204m+20n−8 × 2599mn−20m−55n+4. 4. M1(G,p) = ∑ x1x2∈E(G)P (dx1+dx2 M1(G,p) = ∑ x1x2∈E1(G)P (dx1+dx2) + ∑ x1x2∈E2(G)P (dx1+dx2) ∑ x1x2∈E1(G)P (dx1+dx2) M1(G,p) = ∑ x1x2∈E1(G)P 8 + ∑ x1x2∈E2(G)P 9 + ∑ x1x2∈E1(G)P 10 M1(G,p) = |E1(G)|P8 + |E2(G)|P9 + |E3(G)|P10 M1(G,p) = (10n + 6m + 4)P8 + (4m + 20n− 8)P9 + (99mn− 20m− 55n + 4)P10. 5. M2(G,p) = ∑ x1x2∈E(G)P (dx1+dx2 M2(G,p) = ∑ x1x2∈E1(G)P (dx1×dx2) + ∑ x1x2∈E2(G)P (dx1×dx2 ∑ x1x2∈E1(G)P (dx1×dx2) M2(G,p) = ∑ x1x2∈E1(G)P 16 + ∑ x1x2∈E2(G)P 20 + ∑ x1x2∈E1(G)P 20 M2(G,p) = |E1(G)|P16 + |E2(G)|P20 + |E3(G)|P25 M2(G,p) = (10n + 6m + 4)P16 + (4m + 20n− 8)P20 + (99mn− 20m− 55n + 4)P25. This makes the proof whole. 3. Conclusion and Future Studies In our research article, we investigated indices randic general connectivity index, first zagreb general index, summation general connectivity index, atomic bond connectivity index, geometric arithmetic index, fifth class of geometric arithmetic indices, hyperzagreb index, The initial and secondly multiple [n]-pentacene zagreb indices for a four paraline graphs of these two types of pentacenes. These indices play a crucial role in chemical informatics, specifically in the analysis of organic compounds. The randic index (Rα) is commonly used to explore the physicochemical properties of alkanes, such as boiling point, surface area, and enthalpy of formation. It provides valuable insights into the characteristics of organic molecules. The ABC index is a useful tool for predicting the stability of hydrocarbons, encompassing both linear and branched alkanes. The stability of cycloalkanes can be assessed by the indicated index, which is associated with their strain energy stability. This provides significant insights 10 Int. J. Anal. Appl. (2023), 21:66 into the overall stability of cycloalkanes. In terms of predicting physicochemical characteristics, chem- ical reactivity, and biological activities, the GA index demonstrates superior performance compared to the ABC index. Our investigation of pentacene was approached from a philosophical standpoint rather than relying solely on empirical observations. Our theoretical understanding of pentacenes can sub- stantially benefit in understanding their physical properties, chemical activity and biological activity. A variety from physical feature-related data may be correlated with the chemical structure of pentacenes according to this study’s major results, which may be useful for the power industry. Conflicts of Interest: The authors declare that there are no conflicts of interest regarding the publi- cation of this paper. References [1] X. Li, Y. Shi, A Survey on the Randić Index, MATCH Commun. Math. Comput. Chem. 59 (2008), 127-156. [2] X. Li, Y. Shi, L. Wang, An Updated Survey on the Randić Index, In: I. Gutman, B. Furtula (Eds.), Recent Results in the Theory of Randić Index, University Kragujevac: Kragujevac, Serbia, 2008, pp. 9–47. [3] Z.S. Mufti, S. Zafar, Z. Zahid, M.F. Nadeem, Study of the Paraline Graphs of Certain Benzenoid Structures Using Topological Indices, MAGNT Res. Rep. 4 (2017), 110-116. [4] J. Rada, R. Cruz, Vertex-Degree-Based Topological Indices Over Graphs, MATCH Commun. Math. Comput. Chem. 72 (2014), 603-616 [5] A.M. Hinz, D. Parisse, The Average Eccentricity of Sierpiński Graphs, Graphs Comb. 28 (2011), 671-686. https: //doi.org/10.1007/s00373-011-1076-4. [6] J. Devillers, A.T. Balaban, Topological Indices and Related Descriptors in QSAR and QSPAR, CRC Press, Ams- terdam, 2000. [7] M. Azari, A. Iranmanesh, Harary Index of Some Nano-Structures, MATCH Commun. Math. Comput. Chem. 71 (2014), 373-382. [8] L. Feng, W. Liu, G. Yu, S. Li, The Hyper-Wiener Index of Graphs With Given Bipartition, Utilitas Math. 96 (2014), 99-108. [9] A. Ali, W. Nazeer, M. Munir, S. Min Kang, M-Polynomials And Topological Indices Of Zigzag And Rhombic Benzenoid Systems, Open Chem. 16 (2018), 73-78. https://doi.org/10.1515/chem-2018-0010. [10] M. Knor, B. Luzar, R. Skrekovski, I. Gutman, On Wiener Index of Common Neighborhood Graphs, MATCH Commun. Math. Comput. Chem. 72 (2014), 321-332. [11] K. Xu, M. Liu, K.C. Das, I. Gutman, B. Furtula, A Survey on Graphs Extremal with Respect to Distance-Based Topological Indices, MATCH Commun. Math. Comput. Chem. 71 (2014), 461-508. [12] H.P. Schultz, Topological Organic Chemistry. 1. Graph Theory and Topological Indices of Alkanes, J. Chem. Inf. Comput. Sci. 29 (1989), 227–228. https://doi.org/10.1021/ci00063a012. [13] K. Xu, K.Ch. Das, H. Liu, Some Extremal Results on the Connective Eccentricity Index of Graphs, J. Math. Anal. Appl. 433 (2016), 803-817. https://doi.org/10.1016/j.jmaa.2015.08.027. [14] M.R.R. Kanna, R. Jagadeesh, Topological Indices of Vitamin A, Int. J. Math. Appl. 6 (2018), 271-279. [15] A.U.R. Virk, W. Nazeer, S.M. Kang, On Computational Aspects of Bismuth Tri-Iodide, Preprints, 2018. https: //doi.org/10.20944/preprints201806.0209.v1. [16] M. Dehmer, F. Emmert-Streib, M. Grabner, A Computational Approach to Construct a Multivariate Complete Graph Invariant, Inf. Sci. 260 (2014), 200-208. https://doi.org/10.1016/j.ins.2013.11.008. https://doi.org/10.1007/s00373-011-1076-4 https://doi.org/10.1007/s00373-011-1076-4 https://doi.org/10.1515/chem-2018-0010 https://doi.org/10.1021/ci00063a012 https://doi.org/10.1016/j.jmaa.2015.08.027 https://doi.org/10.20944/preprints201806.0209.v1 https://doi.org/10.20944/preprints201806.0209.v1 https://doi.org/10.1016/j.ins.2013.11.008 Int. J. Anal. Appl. (2023), 21:66 11 [17] L. Feng, W. Liu, A. Ilić, G. Yu, Degree Distance of Unicyclic Graphs with Given Matching Number, Graphs Comb. 29 (2012), 449-462. https://doi.org/10.1007/s00373-012-1143-5. [18] I. Gutman, Selected Properties of the Schultz Molecular Topological Index, J. Chem. Inf. Comput. Sci. 34 (1994), 1087-1089. https://doi.org/10.1021/ci00021a009. [19] M.R. Farahani, M.F. Nadeem, S. Zafar, Z. Zahid, M.N. Husin, Study of the Topological Indices of the Line Graphs of Hpantacenic Nanotubes, New Front. Chem. 26 (2017), 31-38. [20] N. Soleimani, E. Mohseni, N. Maleki, N. Imani, Some Topological Indices of the Family of Nanostructures of Polycyclic Aromatic Hydrocarbons (PAHs), J. Natn. Sci. Found. Sri Lanka. 46 (2018), 81-88. https://doi.org/ 10.4038/jnsfsr.v46i1.8267. [21] N. Soleimani, M.J. Nikmehr, H.A. Tavallaee, Theoretical Study of Nanostructures Using Topological Indices, Stud. U. Babes-Bol, Che. 59 (2014), 139-148. [22] X. Li, H. Zhao, Trees With the First Smallest and Largest Generalized Topological Indices, MATCH Commun. Math. Comput. Chem. 50 (2004), 57–62. [23] B. Zhou, N. Trinajstić, On General Sum-Connectivity Index, J. Math. Chem. 47 (2009), 210-218. https://doi. org/10.1007/s10910-009-9542-4. [24] E. Estrada, L. Torres, L. Rodriguez, I. Gutman, An Atom-Bond Connectivity Index: Modelling the Enthalpy of Formation of Alkanes, Indian J. Chem. 37A (1998), 849-855. [25] D. Vukičević, B. Furtula, Topological Index Based on the Ratios of Geometrical and Arithmetical Means of End-Vertex Degrees of Edges, J. Math. Chem. 46 (2009), 1369-1376. https://doi.org/10.1007/ s10910-009-9520-x. [26] M. Ghorbani, M.A. Hosseinzadeh, Computing ABC4 Index of Nanostar Dendrimers, Optoelectronics Adv. Mater. - Rapid Commun. 4 (2010), 1419-1422. [27] A. Graovac, M. Ghorbani, M.A. Hosseinzadeh, Computing Fifth Geometric-Arithmetic Index for Nanostar Den- drimers, J. Math. Nanosci. 1 (2011), 33-42. https://doi.org/10.22061/jmns.2011.461. [28] P.S. Ranjini, V. Lokesha, M.A. Rajan, on the Schultz Index of the Subdivision Graphs, Adv. Stud. Contemp. Math. 21 (2011), 279-290. [29] P.S. Ranjini, V. Lokesha, I.N. Cangül, On the Zagreb Indices of the Line Graphs of the Subdivision Graphs, Appl. Math. Comput. 218 (2011), 699-702. https://doi.org/10.1016/j.amc.2011.03.125. [30] G. Su, L. Xu, Topological Indices of the Line Graph of Subdivision Graphs and Their Schur-Bounds, Appl. Math. Comput. 253 (2015), 395-401. https://doi.org/10.1016/j.amc.2014.10.053. [31] A. Asghar, A. Qayyum, N. Muhammad, Different Types of Topological Structures by Graphs, Eur. J. Math. Anal. 3 (2022), 3. https://doi.org/10.28924/ada/ma.3.3. [32] Z.H. Niazi, M.A.T. Bhatti, M. Aslam, Y. Qayyum, M. Ibrahim, A. Qayyum, d-Lucky Labeling of Some Special Graphs, Amer. J. Math. Anal. 10 (2022), 3-11. https://doi.org/10.12691/ajma-10-1-2. [33] M. Ahmad, S. Hussain, U. Parveen, I. Zahid, M. Sultan, A. Qayyum, On Degree-Based Topological Indices of Petersen Subdivision Graph, Eur. J. Math. Anal. 3 (2023), 20. https://doi.org/10.28924/ada/ma.3.20. https://doi.org/10.1007/s00373-012-1143-5 https://doi.org/10.1021/ci00021a009 https://doi.org/10.4038/jnsfsr.v46i1.8267 https://doi.org/10.4038/jnsfsr.v46i1.8267 https://doi.org/10.1007/s10910-009-9542-4 https://doi.org/10.1007/s10910-009-9542-4 https://doi.org/10.1007/s10910-009-9520-x https://doi.org/10.1007/s10910-009-9520-x https://doi.org/10.22061/jmns.2011.461 https://doi.org/10.1016/j.amc.2011.03.125 https://doi.org/10.1016/j.amc.2014.10.053 https://doi.org/10.28924/ada/ma.3.3 https://doi.org/10.12691/ajma-10-1-2 https://doi.org/10.28924/ada/ma.3.20 1. Introduction and preliminaries 2. Topological index of four para-line graphs 2.1. Molecular characteristics of the linear [n]-pentacene four para-line graph 2.2. Molecular descriptors of four paraline graphs for multiple pentacenes 3. Conclusion and Future Studies References