On Molecular Descriptors of Carbon Nanocones

Many degree-based topological indices can be obtained from the closed-off M-polynomial of a carbon nanocone. These topological indices are numerical parameters that are associated with a structure and, in combination, determine the properties of the carbon nanocone. In this paper, we compute the closed form of the M-polynomial of generalized carbon nanocone and recover many important degree-based topological indices. We use software Maple 2015 (Maplesoft, Waterloo, ON, Canada) to plot the surfaces and graphs associated with these nanocones, and relate the topological indices to the structure of these nanocones.

devices [17]. More recently, carbon nanocones have gained increased scientific interest due to their unique properties and promising uses in many novel applications such as energy and hydrogen storage. Figures 1 and 2  unique properties and promising uses in many novel applications such as energy and hydrogen storage. Figures 1 and 2 show carbon nanocones.  The molecular graph of   k CNC n nanocones have conical structures 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 the following Figure 3.
In the present report, we give a closed form of the M-polynomial of carbon nanocones. From the M-polynomial, we recover nine degree-based topological indices. In [18], Xu et al. computed the Hosoya polynomial and related distance-based indices for 7 [ ]. In [19], Ghorbani et al. computed the Vertex PI, Szeged, and Omega polynomials of carbon nanocone 4 [ ]. Similarly, many partial results regarding topological indices have been obtained for some particular classes of nanocones. However, we present some general results about complete families of nanocones. Our results present organized generalizations of many existing partial results.    The molecular graph of   k CNC n nanocones have conical structures 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 the following Figure 3.
In the present report, we give a closed form of the M-polynomial of carbon nanocones. From the M-polynomial, we recover nine degree-based topological indices. In [18], Xu  Similarly, many partial results regarding topological indices have been obtained for some particular classes of nanocones. However, we present some general results about complete families of nanocones. Our results present organized generalizations of many existing partial results. The molecular graph of CNC k [n] nanocones have conical structures 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 the following Figure 3.
In the present report, we give a closed form of the M-polynomial of carbon nanocones. From the M-polynomial, we recover nine degree-based topological indices. In [18]
Throughout this paper, G denotes connected graph, V(G) and E(G) denote the vertex set and the edge set, respectively, and denotes the degree of a vertex.

Definition 1.
The M-polynomial of G is defined as: The first well-known topological index was introduced by Wiener [21] when he was studying the boiling point of paraffin. He named it the path number, which is now known as the Wiener index [22,23]. Later, Randic defined the first degree-based topological index in 1975 [24]. The Randic index is denoted by 1/2 () RG  , and is defined as: In 1998, working independently, Bollobas and Erdos [25] and Amic et al. [26] proposed the generalized Randic index, which has been studied extensively by both chemists and mathematicians [27]. Many mathematical properties of the Randic index have been discussed [28]. For a detailed survey we refer to the monograph of Li and Gutman [29].
The general Randic index is defined as:
Throughout this paper, G denotes connected graph, V(G) and E(G) denote the vertex set and the edge set, respectively, and d v denotes the degree of a vertex. Definition 1. The M-polynomial of G is defined as: The first well-known topological index was introduced by Wiener [21] when he was studying the boiling point of paraffin. He named it the path number, which is now known as the Wiener index [22,23]. Later, Randic defined the first degree-based topological index in 1975 [24]. The Randic index is denoted by R −1/2 (G), and is defined as: In 1998, working independently, Bollobas and Erdos [25] and Amic et al. [26] proposed the generalized Randic index, which has been studied extensively by both chemists and mathematicians [27]. Many mathematical properties of the Randic index have been discussed [28]. For a detailed survey we refer to the monograph of Li and Gutman [29].
The general Randic index is defined as: Gutman and Trinajstic introduced the first Zagreb index and second Zagreb index, which are defined as: modified Zagreb index was defined as: .
The symmetric division index is defined as: Other well-known topological indices are the harmonic index, and the augmented Zagreb index [34,35]: The following Table 1 relates some well-known degree-based topological indices with the M-polynomial [5]. Table 1. Derivation of some degree-based topological indices from the M-polynomial.

Topological Index
Derivation from M(G; x, y) where:

Results
In this section, we give our computational results.
Proof. Let CNC k [n] be the graph of nanocones. From the graph of CNC k [n] nanocones, we can see that there are two partitions, The edge set of the CNC k [n] can be partitions as follows: and:
Now we recover degree-based topological indices by using Table 1. The Figures 5-13 show the relations of different topological indices with values of k and n. It is noticeable that all of the above discussed topological indices vary quadratically with n, and linearly with k.  3. m M 2 (CNC k [n]) = S x S y ( f (x, y)) x=y=1 = 1 9 kn 2 + 10 27 kn + 1 4 k.   6. SDD(CNC k [n]) = D x S y + D y S x ( f (x, y)) x=y=1 = 2kn 2 + 5kn + 2k.

Conclusions
The closed form of the M-polynomials of all of the carbon nanocones is computed. This polynomial generates a lot of information about degree-based topological descriptors, which are actually graph invariants. These indices, in combination, determine the properties of nanocones. The topological indices calculated in this paper are important for guessing the physicochemical properties of understudy chemical compounds. For example, the Randić index is a topological descriptor that Figure 11. Plots for the harmonic index (left for arbitrary n and k, middle for k = 4, and right for n = 5). 8. I(CNC k [n]) = S x JD x D y ( f (x, y)) x=1 = 3 2 kn 2 + 29 10 kn + k.

Conclusions
The closed form of the M-polynomials of all of the carbon nanocones is computed. This polynomial generates a lot of information about degree-based topological descriptors, which are actually graph invariants. These indices, in combination, determine the properties of nanocones. The topological indices calculated in this paper are important for guessing the physicochemical properties of understudy chemical compounds. For example, the Randić index is a topological descriptor that

Conclusions
The closed form of the M-polynomials of all of the carbon nanocones is computed. This polynomial generates a lot of information about degree-based topological descriptors, which are actually graph invariants. These indices, in combination, determine the properties of nanocones. The topological indices calculated in this paper are important for guessing the physicochemical properties of understudy chemical compounds. For example, the Randić index is a topological descriptor that

Conclusions
The closed form of the M-polynomials of all of the carbon nanocones is computed. This polynomial generates a lot of information about degree-based topological descriptors, which are actually graph invariants. These indices, in combination, determine the properties of nanocones. The topological indices calculated in this paper are important for guessing the physicochemical properties of understudy chemical compounds. For example, the Randić index is a topological descriptor that has been connected with numerous substance qualities of atoms, and has been found to be parallel to processing the boiling point and Kovats constants of the particles. To associate with certain physicochemical properties, the GA index has a very preferable prescient control over the prescient intensity of the Randić index. The first and second Zagreb indexes were found to calculate the aggregate π-electron vitality of the atoms inside specific surmised articulations. These are among the graph invariants, which were proposed for the estimation of the skeleton of the spreading of the carbon molecule. To calculate the distance-based topological indices of understudied nanocones is an interesting problem that is worthy of further investigation.