Computational aspects of line graph of carbon nanocones

Zafar Hussain1, Mobeen Munir2, Waqas Nazeer3, Muhammad Shoaib Saleem4, Shin Min Kang5,6* and Young Chel Kwun7 1 Department of Mathematics and Statistics, University of Lahore, Lahore, Pakistan. 2 Department of Mathematics Division of Science and Technology, University of Education, Lahore, 54000, Pakistan. 3 Department of Mathematics, Government College University, Lahore, Pakistan. 4 Division of Science and Technology, University of Okara, Pakistan. 5 Department of Mathematics and RINS, Gyeongsang National University, Jinju, South Korea. 6 Center for General Education, China Medical University, Taiwan, Taichung, Taiwan 7 Department of Mathematics, Dong-A University, Busan, South Korea.


INTRODUCTION
In mathematical chemistry, mathematical tools such as polynomials and numbers predict properties of compounds without using quantum mechanics. These tools, in combination, capture information hidden in the symmetry of molecular graphs. Topological indices are numerical parameters of a graph, which characterise its topology and are usually graph invariant. They describe the structure of molecules numerically and are used in the development of qualitative structure activity relationships (QSARs). The most commonly known invariants of such kind are degree-based topological indices. These are actually the numerical values that correlate the structure with various physical properties, chemical reactivity, and biological activities (Klavžar & Gutman, 1996;Rucker & Rucker, 1999;Brückler et al., 2011). It is an established fact that many properties such as heat of formation, boiling point, strain energy, rigidity and fracture toughness of a molecule are strongly connected to its graphical structure and this fact plays a synergic role in chemical graph theory.
Several algebraic polynomials have useful applications in chemistry such as Hosoya polynomial (also called Wiener polynomial) (Gutman, 1993),

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Journal of the National Science Foundation of Sri Lanka 47 (4) which play a vital role in determining distance-based topological indices. Among other algebraic polynomials, M-polynomial (Deutsch & Klavzar, 2015) introduced in 2015, plays the same role in determining the closed form of many degree-based topological indices (Munir et al. 2016a(Munir et al. , b, c, d, 2017Kwun et al., 2017;Ali et al., 2018).
The main advantage of M-polynomial is the wealth of information that it contains about degree-based graph invariants.
Carbon nanocones have been observed since 1968 or even earlier (Gillot et al., 1968) on the surface of naturally occurring graphite. Their bases are attached to the graphite and the height varies between 1 to 40 micrometers. The walls are often curved and are less regular than those of the laboratory made nanocones. Carbon nanostructures have attracted considerable attention due to their potential uses in many applications including energy storage, gas sensors, biosensors, nanoelectronic devices and chemical probes (Iijima, 1991). Carbon allotropes such as carbon nanocones and carbon nanotubes have been proposed as possible molecular gas storage devices (Zhao et al., 2002;Adisa et al., 2011). More recently, carbon nanocones have gained increased scientific interests due to their unique properties and promising uses in many novel applications such as energy and gas storage. Figure 1 depicts the structure of a carbon nanocone.
The molecular graph of [ ] k CNC n nanocones has a conical structure with a cycle of length k at its core and n layers of hexagons placed at the conical surface around its centre as shown in Figure 2. possible molecular gas storage devices (Adisa, et al., 2011;Zhao, et al., 2002). More recently, carbon nanocones have gained increased scientific interests due to their unique properties and promising uses in many novel applications such as energy and gas storage.  The molecular graph of   k CNC n nanocones has a conical structure with a cycle of length k at its core and n layers of hexagons placed at the conical surface around its center as shown in following figure 2. The line graph L(G) of a graph G is the graph each of whose vertex represents an edge of G and two of its vertices are adjacent if their corresponding edges are adjacent in G ( Huo, et al., 2016;Iranmanesh, et al., 2009). In the following figure 3, we can see the graph of the carbon nanocones   k CNC n The present paper is concerned with the line graph of the carbon nanocone   k CNC n where k is the number of hexagons forming the base of the cone and n is the number of layers of the cone. In (Wu, et al., 2010) authors computed the Wiener index of a graph defined as The line graph L(G) of a graph G is the graph each of whose vertex represents an edge of G and two of its vertices are adjacent if their corresponding edges are adjacent in G ( Huo, et al., 2016;Iranmanesh, et al., 2009). In the following figure 3, we can see the graph of the carbon nanocones   k CNC n The present paper is concerned with the line graph of the carbon nanocone   k CNC n where k is the number of hexagons forming the base of the cone and n is the number of layers of the cone. In (Wu, et al., 2010) authors computed the Wiener index of a graph defined as The line graph L(G) of a graph G is the graph each of whose vertices represents an edge of G and two of its vertices are adjacent if their corresponding edges are adjacent in G (Iranmanesh et al., 2009;Huo et al., 2016). In Figure 3 The present paper is concerned with the line graph of the carbon nanocone [ ] k CNC n where k is the number of hexagons forming the base of the cone and n is the number of layers of the cone. In Wu et al. (2010), authors computed the Wiener index of a graph defined as . They also computed the Wiener index of the line graph and established new bounds on the Wiener index of the line graph of any graph. In Buckley and Rapoport (1981), authors computed mean distances in line graphs. In Cataldo et al. (2010), Wiener index was used as pure topological potential energy of graphenic infinite structures producing interesting results, assigning to fullerene-like nanostructures a nonnegligible probability of formation on graphene sheets on carbon black or on amorphous carbon in general.
In this paper, we compute M-polynomial and degreebased topological indices of the line graph of nanocones. We also give many degree-based topological indices in closed forms for the line graphs of nanocones.

METHODOLOGY, RESULTS AND DISCUSSION
Throughout this paper, G is a connected graph, V(G) and E(G) are the vertex set and edge set, respectively and v d denotes the degree of a vertex v. Definition 1. (Deutsch & Klavzar, 2015). The M-polynomial of G is defined as: The first topological index was introduced by Wiener (1947) and it was named path number, which is now known as Wiener index. In chemical graph theory, this is the most studied molecular topological index due to its wide applications [see for details (Gutman & Polansky, 1986;Dobrynin et al., 2001)]. Randic´ index, (Randić, 1975) denoted by 1/2 ( ) R G − and introduced by Milan Randić in 1975 is also one of the oldest topological indices. The Randić index is defined as: Working independently, Bollobas and Erdos (1998) and Amić et al. (1998) proposed the generalised Randic´ index and it has been studied extensively by both chemists and mathematicians (Hu et al., 2005). Many mathematical properties have been discussed (Caporossi et al., 2003). For a detailed survey, we refer the book (Li et al., 2006).
The general Randić index is defined as: Here α can be any The Randić index is the most popular, most often applied and most studied among all other topological indices. Many papers and books such as Kier and Hall (1976; are written on this topological index. Randić himself wrote two reviews on his Randic index (Randić, 2001;2008) and there are more reviews (for example Li & Shi, 2008). The suitability of the Randić index for drug design was immediately recognised, and eventually, the index was used for this purpose on countless occasions. The physical reason for the success of such a simple graph invariant is still an enigma, although several moreor-less plausible explanations were offered. Gutman and Trinajstic´ (1972) introduced the first Zagreb index and second Zagreb index, which are defined as: modified Zagreb index is defined as: For details about these indices, we refer Nikolić et al., (2003), Gutman and Das (2004) and Zhou and Trinajstić (2010) to the readers.
The symmetric division index is defined as:

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Journal of the National Science Foundation of Sri Lanka 47 (4) Another variant of Randic index is the harmonic index defined as: The augmented Zagreb index is defined as: and it is useful for computing heat of formation of alkanes.
[For details about these topological indices we refer (Rehman et al., 2017;Sardar et al., 2017;De, 2018)]. Table 1 relates some well-known degree-based topological indices with M-polynomial (Deutsch & Klavzar, 2015).                    L CNC n . These indices are actually, experimentally correlated with many properties. All topological indices calculated depend similarly on k and n. Figure 5 is the surface associated to topological index, whereas Figure  6 shows that topological indices are quadratically related with n, but are linearly related with k as Figure 7 suggests.
We pose an open problem to compute the Wiener and other distance-based indices of carbon general nanocones and its line graph. Although some partial results are available for one particular subclass of these nanocones it will be interesting to see how the reactivity is affected by the change in sizes of general nanocones and their Weiner indices.