M-Polynomial and Degree Based Topological Indices of Some Nanostructures

: The association of M-polynomial to chemical compounds and chemical networks is a relatively new idea, and it gives good results about the topological indices. These results are then used to correlate the chemical compounds and chemical networks with their chemical properties and bioactivities. In this paper, an effort is made to compute the general form of the M-polynomials for two classes of dendrimer nanostars and four types of nanotubes. These nanotubes have very nice symmetries in their structural representations, which have been used to determine the corresponding M-polynomials. Furthermore, by using the general form of M-polynomial of these nanostructures, some degree-based topological indices have been computed. In the end, the graphical representation of the M-polynomials is shown, and a detailed comparison between the obtained topological indices for aforementioned chemical structures is discussed.


Introduction and Preliminaries
A molecular graph (in chemical graph theory) is a simple graph (a graph that does not contain edges whose two ends are the same vertex, nor containing two edges with the same two ends) where the atoms of the molecule are represented as vertices and the bonds between them are the edges. A graph G(V, E) consists of a vertex set V(G) and an edge set E(G). The number of edges incident to v of some vertex of a graph G is said to be its degree and is denoted by d v ; δ is the minimum degree of a graph, and ∆ is its maximum. We can represent an edge with vertices, e = uv ∈ E(G), if it connects vertex u to vertex v.
Topological indices are a numerical quantity computed from the molecular graph of a chemical compound and typically remain the same under graph isomorphism. A topological index determined by means of using the degrees of the molecular graph is a family of indices that can be used to point out and model certain properties of chemical compounds; these indices are also known as degree-based topological indices [1][2][3][4]. In the same context, the M-polynomial plays a similar part; we can obtain closed-form formulas for degree-based topological indices from it [5][6][7][8][9][10].
There are some recent papers devoted to the computation of the M-polynomial of a given chemical graph. In particular, in [11], the authors considered two infinite classes NS1[n] and NS2[n] of nanostar dendrimers and computed the Zagreb indices and Zagreb polynomials for these special nanostar dendrimers. In [12], the author determined certain Zagreb polynomials of a special type of dendrimer nanostar D 3 [n], and the Zagreb indices and Zagreb polynomials of the same dendrimer nanostars were computed in [13]. The computation of some fifth multiplicative Zagreb indices of PAMAMdendrimers was reported in [14]. The first and second reverse Zagreb indices, reverse hyper-Zagreb indices, and their polynomials of porphyrin, propyl ether imine, zinc porphyrin, and poly(ethylene amido amine) dendrimers were determined in [15]. The computation of the M-polynomial and topological indices of the benzene ring embedded in a P-type surface network was given in [16]. The M-polynomial and degree-based topological indices of polyhex nanotubes were given in [9]. The main object of this paper is to derive the general form of the M-polynomial for two dendrimer nanostars ND 1 [n], ND 2 [n] ( Figure 1A,B) and four types of nanotubes. Furthermore, we compute the first and second Zagreb index, the modified Zagreb index, the Randic index, and the symmetric division index for this family of molecules. We also give the graphical representation of the M-polynomial. A comparison between the obtained topological indices of these chemical graphs is outlined at the end.

Materials and Methods
Dendrimers are repetitively branching artificial star-shaped molecules synthesized from branched units called monomers using a nanoscale fabrication process; they have a well-defined structure with three major components: the core, branches, and end groups, where new branches are made from the core and are added recursively in steps. There are various applications of nanostar dendrimers for the formation of micro-and macro-capsules, chemical sensors, colored glasses, and modified electrodes. Due to these applications, these compounds have attracted much attention in both chemistry and mathematics alike.
Nanotubes are a nanoscale material that takes up a tube-like structure and can be made from various materials, organic or inorganic, such as carbon, boron, and silicon. CNT (Carbon Nanotubes) have a high thermal and electrical conductivity, while also being highly flexible and being good electron field emitters. They have various applications in energy storage, air and water filtration, fabrics, and thermal compounds and have attracted much attention from the scientific community.
Boron nanotubes are attracting much attention as well because of their remarkable properties such as structural stability, work function, transport properties, and electronic structure and are coming to challenge CNTs. In this article, we study the structures of two configurations of CNT, "zig-zag" G 1 [n, m] and "armchair" G 2 [n, m] (Figure 2A  As introduced by Grutman and Trinajstic, the first Zagreb index M 1 (G) and the second Zagreb index M 2 (G) are defined as follows [17]: It is well known that the first Zagreb index can be written as: The modified Zagreb index m M 2 (G) is defined to be: , the general Randic index, is defined to be: The symmetric division index, SDD(G), is defined to be: Let G be a simple molecular connected graph, and let d w denote the degree of some vertex v ∈ V(G). The vertex set V(G) can be partitioned as follows: The edge set E(G) can be partitioned into the following sets: Then, the M-polynomial of a graph G denoted as M(G, x, y) is defined to be: From the above polynomial, one can obtain numerous degree-based topological indices by differentiating and/or integrating with respect to x and y [6]. The Table 1 below lists some of the degree-based topological indices and how to derived them from the M-polynomial. Table 1. Derivation of degree-based topological indices form M-polynomial.

Topological Index g(x, y)
Derivation from M(G, x, y) where D x , D y , S x , S y is defined to be:

Results
We found the M-polynomials of two special families of dendrimer nanostars denoted by ND 1 [n] andND 2 [n] and four types of nanotubes denoted by G k [n, m] (k = 1, 2, 3, 4). After that, we represented some of the M-polynomials as three-dimensional surfaces using Mathematica, then with the assistance of the above table, we found the topological indices of the aforementioned structures.

M-Polynomials of the Dendrimer Nanostars
The main objective of this subsection is to obtain the general closed form of the M-polynomial associated with the two special nanostar dendrimers (ND 1 [n] & ND 2 [n]). Theorem 1. Let ND 1 [n] be the molecular graph whose structure can be seen in Figure 1A. Then, its M-polynomial is: Proof. Studying the construction of ND 1 [n] (see Figure 1A), we can see that there are two partitions of the vertex set of ND 1 [n] as: We also see that E(ND 1 [n]) can be partitioned as follows: Thus, following from the definition, the M-polynomial of ND 1 [n] is: be the molecular graph whose structure can be seen in Figure 1B. Then, its M-polynomial is: Proof. Studying the construction of ND 2 [n] (see Figure 2B), we can see that there are two partitions of the vertex set of ND 2 [n] as: We also see that E(ND 2 [n]) can be partitioned as follows: Thus, following straight from the definition, the M-polynomial of ND 2 [n] is:

M-Polynomials of the Nanotubes
In this subsection, we will provide the general formula for the M-polynomial of the four nanotubes (G k [n, m], where k = 1, 2, 3, 4), as shown in Figures 2A,B and 3A,B. We also provide the graphical representation of these M-polynomials. Theorem 3. Let G 1 [n, m] be a molecular graph whose structure can be seen in Figure 2A. Then: Proof. Studying the construction of G 1 [n, m], we can see that there are two partitions of the vertex set of G 1 [n, m] as: We also see that E(G 1 [n, m]) can be partitioned as follows: . Thus, following straight from the definition, the M-polynomial of G 1 [n, m] is: Theorem 4. Let G 2 [n, m] be a molecular graph whose structure can be seen in Figure 2B. Then: Proof. Studying the construction of G 2 [n, m], we can see that there are two partitions of the vertex set of G 2 [n, m]: We also see that E(G 2 [n, m]) can be partitioned as follows: = 2mx 4 y 4 + 4m x 4 y 6 + (3mn − 8m)x 6 y 6 Theorem 6. Let G 4 [n, m] be a molecular graph whose structure can be seen in Figure 3B We also see that E(G 4 [n, m]) can be partitioned as follows: Now, we will give the 3D-plot of some of these M-polynomials with the help of Mathematica. In particulary, the 3d-plot of M-polynomial of dendrimer nanostar ND 1 [n] is shown in Figure 4 and for two nanotubes G 3 [n, m] and G 4 [n, m] in Figures 5 and 6 respectively.
On the testing sets supplied by the International Academy of Mathematical Chemistry, it showed good predictive properties. It can also decently predict the total surface area for polychlorobiphenyls. Therefore, from Figure 8, we note that the symmetric division index of ND 2 [n] ND 1 [n] rises sharply with ND 2 [n, m] being much greater than ND 1 [n, m]. Similarly, we can also take note in reference to

Conclusions
We computed the M-polynomial of two classes of nanostar dendrimers and four nanotubes and used the M-polynomials to obtain a wide list of indices for our molecular graphs. By analyzing our results, we found that ND 2 [n] were better at achieving larger values of the calculated topological indices in comparison to ND 1 [n] and likewise that G 3 [n, m] were better at achieving larger values of these indices than the other three nanotubes. For future prospects, one can compute the M-polynomial for other types of nanostar dendrimers or nanotubes. For example, single-walled inorganic aluminosilicate and aluminogermanate nanotubes of a periodic structure and precise dimensions can be investigated related to the current study.