#### Abstract

The topological invariants are related to the molecular graph of the chemical structure and are numerical numbers that help us to understand the topology of the concerned chemical structure. With the help of these numbers, many properties of graphene can be guessed without preforming any experiment. Huge amount of calculations are required to obtain topological invariants for graphene, but by applying basic calculus roles, neighborhood -polynomial of graphene gives its indices. The aim of this work is to compute neighborhood degree-dependent indices for the graph of graphene and the line graph of subdivision graph of graphene. Firstly, we establish neighborhood *M*-polynomial of these families of graphs, and then, by applying basic calculus, we obtain several neighborhood degree-dependent indices. Our results play an important role to understand graphene and enhance its abilities.

#### 1. Introduction

Graph is the union of edges (lines) and vertices (nodes), and the study of graphs is known as the graph theory [1, 2]. In this study, we study the properties of graphs and invent applications of different families of graphs [3, 4]. Today, the graph theory got applications in almost all areas of sciences and engineering, and applying the graph theory to solve problems of chemistry is known as the chemical graph theory [5, 6].

In the chemical graph theory, topological indices are graph invariants that remain the same up to graph isomorphism and help us to attain properties of molecular graphs without any lab work. The first topological index was introduced by Wiener in 1947 when he was working on the boiling point of alkane [7]. Today, this index is named the Wiener index, which is a distance-based topological index. After Wiener, Randić introduced the first degree-depend index, which was firstly named the path index, but today, it is known as the Randić index [8]. This index got huge attention of researchers due to its amazing applications. After the promising success of the Randić index, Gutman coined the idea of Zagreb indices and many research papers are written on this index (see [9] and references therein).

Since now, a single index can give all the information about any molecular graph, so till now, more than 150 indices are defined and studied [10–12]. Since huge amount of work is required to obtain topological indices, so the research found a shortcut to obtain them and introduced *M*-polynomial [13] and this polynomial gives easy way to compute almost all degree-based indices [14, 15]. Many studies have been performed so far on *M*-polynomial [16, 17], and still, there are many known interesting macular graphs whose *M*-polynomial can be established. Following *M*-polynomial, neighborhood -polynomial was introduced in Ref. [18], which is helpful in determining neighborhood degree sum-based graphical indices.

The aim of this work is to compute neighborhood degree-dependent indices for the graph of graphene and the line graph of subdivision graph of graphene. Firstly, we establish neighborhood *M*-polynomial of these families of graphs, and then, by applying basic calculus, we obtain several neighborhood degree-dependent indices.

#### 2. Primaries

Throughout this work, we consider only finite, simple, undirected, and connected graphs with a vertex set and an edge set [19, 20]. The degree of a vertex is the number of vertices adjacent to [21, 22]. The neighborhood degree sum-based graphical indices depend upon the neighborhood degree, which is defined by for an edge , where and are vertices of the edge . Neighborhood degree sum [23, 24] of a vertex is defined as the sum of degrees of all vertices that are adjacent to the vertex. The degree of a vertex is the total number of edges incident to the vertex. The line graph of a graph is the graph whose vertex set corresponds to the edges of such that two vertices of are adjacent if the corresponding edges of are adjacent (see for details [25, 26] and references therein). The subdivision graph of a graph is the graph obtained from by replacing each of its edges by a path of length two (see [27, 28]).

The neighborhood -polynomial of a graph is defined as [18]where is the total number of edges such that . We use for in this work.

The neighborhood degree-based graphical indices defined on the edge set of a graph can be expressed aswhere is the function of used in the definition of neighborhood degree-based indices. The above result can also be written as

By taking in equation (2), we get the third version of the Zagreb index, the neighborhood second Zagreb index, the neighborhood forgotten graphical index, the neighborhood second modified Zagreb index, the neighborhood general Randić index, the third index, the fifth index, the neighborhood Harmonic index, the neighborhood inverse sum index, the Sanskruti index, the fifth hyper Zagreb index, the fifth hyper Zagreb index, the fifth arithmetic-geometric index, and the fifth geometric-arithmetic index, respectively [29–32].

All the abovementioned indices can be computed directly from *NM*-polynomial, and the relations of some neighborhood degree-based graphical indices with the -polynomial are shown in Table 1, where , , , , and , are the operators.

#### 3. Neighborhood -Polynomial of Graphene Networks

In this section, we find neighborhood -polynomial of graphene and calculate neighborhood degree-based graphical indices of graphene by using its neighborhood -polynomial.

##### 3.1. Graphene Networks

Graphene is an atomic scale honeycomb lattice made of the carbon atoms. Graphene is denoted by , where is the number of rows of benzene rings and is the number of benzene rings in each row. Graphene has vertices and edges [33]. Figure 1 shows the graphene .

Theorem 1. *Let ** be a graphene network, with ** and ** ; then, its neighborhood ** -polynomial is given by*

*Proof. *Graphene has vertices and edges. The edge set of , for and , can be partitioned as follows:Thus, the neighborhood -polynomial of , for and , isThis is the required neighborhood -polynomial of , for and .

Theorem 2. *Let ** be a graphene network, with ** and ** ; then, its neighborhood ** -polynomial is given by*

*Proof. *Graphene has vertices and edges. The edge set of , for and , can be partitioned as follows:Thus, the neighborhood -polynomial of , for and , isThis is the required neighborhood -polynomial of , for and .

Theorem 3. *Let ** be a graphene network, with ** and ** ; then, its neighborhood ** -polynomial is given by*

*Proof. *Graphene has vertices and edges. The edge set of , for and , can be partitioned as follows:Thus, the neighborhood -polynomial of , for and , isthis is the required neighborhood -polynomial of , for and .

Now, we calculate neighborhood degree-based graphical indices of graphene by using its neighborhood -polynomial.

Corollary 1. * Let be a graphene network, with and ; then, its neighborhood degree-based graphical indices are given by*(1)

*(2)*

*(3)*

*(4)*

*(5)*

*(6)*

*(7)*

*(8)*

*(9)*

*(10)*

*(11)*

*(12)*

*(13)*

*(14)*

*Proof. * -Polynomial of , for and , is given byLetThen, we have(1) is defined as Now, by using equation (15), we have(2) is defined as Now, by using equation (15), we have(3) is defined as Now, by using equation (15), we have(4) is defined as Now, by using equation (15), we have(5) is defined as Now, by using equation (15), we have(6) is defined as Now, by using equation (15), we have(7) is defined as Now, by using equation (15), we have(8) is defined as Now, by using equation (15), we have(9) is defined as Now, by using equation (15), we have(10) is defined as Now, by using equation (15), we have(11)Fifth hyper Zagreb index is defined as Now, by using equation (15), we have(12)Fifth hyper Zagreb index is defined as Now, by using equation (15), we have(13)Fifth arithmetic-geometric index is defined as Now, by using equation (15), we have