Computing Topological Indices for Para-Line Graphs of Anthracene

Abstract Atoms displayed as vertices and bonds can be shown by edges on a molecular graph. For such graphs we can find the indices showing their bioactivity as well as their physio-chemical properties such as the molar refraction, molar volume, chromatographic behavior, heat of atomization, heat of vaporization, magnetic susceptibility, and the partition coefficient. Today, industry is flourishing because of the interdisciplinary study of different disciplines. This provides a way to understand the application of different disciplines. Chemical graph theory is a mixture of chemistry and mathematics, which plays an important role in chemical graph theory. Chemistry provides a chemical compound, and graph theory transforms this chemical compound into a molecular graphwhich further is studied by different aspects such as topological indices.We will investigate some indices of the line graph of the subdivided graph (para-line graph) of linear-[s] Anthracene and multiple Anthracene.


Introduction
Chemical graph theory is a branch of mathematical chemistry that is concerned with analyses of all consequences of a connectivity in a chemical graph. Some physical properties, e.g., breaking point, can be anticipated in view of the structure of the atoms. Numerical and computational systems are viably used to show and predict the structure at an atomic level. The structures of atoms, from anumerical perspective, are graphs. Graph theory is utilized as a part of nearly every field of science, and it is likewise utilized for training, both for recreation and design.
V(G) and E(G) compose the vertex set and edge set of a graph Grespectively; p, q ∈ V(G) are adjacent if p and q are end points of u ∈ E(G)" and u is an edge whose end vertices are p and q. The set of all neighbors of a vertex prepresented by Np, is called the neighborhood of p. The count of edges that occur on a vertex is called the degree of the vertex denoted by ξ p and Sp = ∑︀ q∈Np ξ p , where Np = {q ∈ V(G) : pq ∈ E(G)}. We can construct the line graph L(G) of any graph G in such a way that the edges of the original graph will become the vertices of line graph i.e. two vertices p and q occur if and only when these vertices have a common end vertex in G. The para-line graph of Gis the line graph of the subdivision of Gi.e. L(S(G)), which will be represented as G * . The subdivision graph is the graph attained from G by replacing each of its edges by a path of length 2. For instance, consider C2H6 as the hydrocarbon named Ethane which is characterized as a molecular structure. The graph of C2H6 is shown in The topological index which is also referred to as the molecular descriptor, is a real number which describes the properties of a certain chemical compound. The study of topological indices on different chemical structures has been an area of research for all graph theorists. It is a bridge between mathematics and chemistry. The molecular descriptors which are separated into three groups depend on degree-based [16,24], distance-based [8,29] and spectrum-based [2,18,20,21] indices. Some other topological indices have also been studied which are centred on both degrees and distances [4,7,11,19].
The 1 st general Zagreb index in [17] is the oldest and most used molecular descriptor and is defined as: The general sum-connectivity index χ α (G) is defined as [30]: The ABC is specified by Estrada in [5]. The ABCindex of graph Gis defined as: ABC 4 index was presented by Ghorbani in [9] is explained as: Rα (general Randic connectivity index) of G is proposed as [1]: Where ∝ ∈ R. If α is −0.5, then R −0.5 (G) is called Randic connectivity index of G. Vukicevic and Furtula presented the (GA) index in [28]. The GAindex for graph G is defined as: GA 5 was introduced by Graovac et al. in [10] is proposed as: The hyper-Zagreb index is suggested as: In 2012, Ghorbani and Azimi introduced the Zagreb indices in different form as, the 1 st and 2 nd multiple Zagreb index PM 1 (G)and PM 2 (G), 1 st and 2 nd Zagreb polynomial M 1 (G, a) and M 2 (G, a) respectively, are suggested as:

Applications of Topological Indices
Randic observed the correlation between the Randic index and physio-chemical properties of alkane such as boiling point, entholphies of formation, surface area and so on. The ABC index is a very effective index in heat formation [5]. GA index is a better pridictive index than the Randic Index as GA has much chosen prophetic control on the prophetic energy of the Rα [3]. The 1 st and 2 nd Zagreb index were very useful in the calculation of the aggregate π-electron energy of the molecule [12]. These molecular descriptors were suggested for the approximation of streched carbon-skeleton [13].

Topological indices of para-line graphs
Para-line graphs are an attractive field of study in chemical graph theory. Ranjini et al. calculated the explicit expression for the Schultz index and Zagreb index of the para-line graphs of the wheel, ladder, helm, tadpole [25,26]. In 2015, Su et al. [27] investigated χ α (general sumconnectivity index )and co-index of the above mentioned graph of wheel, tadpole. To study the para-line graphs S (8,9) 16s − 8 7 S (9,9) 10s − 8 of ladder, tadpole and wheel, Nadeem et al. in [22] evaluated the ABC 4 , GA 5 index and calculated the generalized Randic index, general Zagreb index, χ α , ABC index, GA index, ABC 4 index and GA 5 index of TUC 4 C 8 [p, q] in [23]. Klein et al. [15] offered few applications and basic properties of the para-line graphs in chemical graph theory. Gutman also shed a light on the application of line graphs see [14]. Estrada showed the application of line graph in [6].

Results for para-line graph of linear [s]-Anthracene
In Figure 2, the graph of linear [s]-Anthracene is presented and it is denoted by Ts. Ts has 14s vertices and 18s − 2 edges. Suppose the graph of Anthracene contains three hexagon which are connected with a square expands vertically and horizontally. Let the edge e (p,q) represent the count of edges connecting the vertices of degree ξ p and ξ q . The graph of Anthracene holds the following edges shown in Table 2: Proof. In G * , the overall count of the vertices is 48s − 10 which is the sum of degree two and degree three vertices 8 + 12s, 24s − 12respectively. As
Using Table 1, we obtained the following results.

Conclusion
In this paper, we have comprehended the work on these indices: Rα, "Mα, χ α , ABC, GA, ABC 4 , GA 5 , PM 1 , PM 2 , M 1 (G, a), and M 1 (G, a) of the line graph of the subdivided graph( para-line graph) of linear [s]-Anthracene and multiple Anthracene. Randic index Rα plays a vital role in the ex-tension of the branching of the carbon-atom skeleton of hydrocarbons. PM 1 (G) and PM 2 (G) may be expressed in the QSPR study and shows a central role in the analysis of the boiling and melting point of drugs. Chemical graphs are currently being studied with the help of para-line graphs which are very important in the field of chemistry. Our upcoming work will be the emphasis on some new classes of the above mentioned graphs(para-line) of chemical structures with respect to the different topological indices.
Ethical approval: The conducted research is not related to either human or animal use.

Conflict of Interests:
Authors declare no conflict of interest.