#### Abstract

The concept of M-polynomials determines the algebraic form of a system or network. It creates a structure into an algebraic equation and makes work easy to do on such a structure. This has diverse uses in different applied mathematics and as well as in engineering fields. In this study, we look closely at the abstract form of *Y*-shaped junctions. For the generic view of *Y*-shaped junctions, we developed some vertex-degree-based M-polynomials formulas. On *Y*-shaped junctions, we discussed some topological index-based concepts as well and verified the results available in the literature.

#### 1. Introduction

The future of multiterminal networks and electronic appliances is possible by the building blocks made by nanotube junctions or branched nanotubes. There are many types of nanotube junctions serving nano-electronics, such as *L*, *T*, *Y*, and *X*. For the literature on the electrical properties of different junctions, we refer to see [1–3], its mechanical stability [4], conduction mechanism and transport phenomena [5], thermal rectification [6]. Instead of straight nanotubes, nanostructures developed using the nanotube junctions are anticipated to deliver much better in terms of mechanical properties. In order to create nanostructures, nanotubes are combined, their techniques, approaches, and succinct descriptions are available (see, for example, [7, 8]).

In this presented work, we will only consider discussing *Y*-type junctions of carbon nanotubes. It is obvious by name that *Y*-junctions are having the shape of *Y*-alphabet and are made by joining three nanotubes. These three nanotubes are joined on a point named as branching point as shown in Figure 1, which is itself comprised of hexagons and six heptagons. If the pattern of hexagons surrounds each heptagon with a pattern distributed symmetrically, and nanotubes are joined in the identical *Y*-shape, these types of *Y*-junctions are symmetric.

On the topology of nanotubes, *Y*-junctions may have further subtypes. For example, single-walled nanotubes have three types of geometries, chiral, zig-zag, and armchair. So developed *Y*-junctions are regarded as armchair carbon nanotube *Y*-junction. All these properties are available for multiwalled nanotubes. In 1991, the first nanotube was discovered [9], and after two years, the first structural topology of a *Y*-shaped junction was made by the authors of [10, 11] independently. The researchers in [12] experimentally observed these junctions in 1995. In [13, 14], the authors presented the idea of asymmetric and symmetric *Y*-shaped junctions, their constructions, their models, and their numerous variants. The applications and properties of novel structure *Y*-shaped junctions are presented in [15, 16].

The quantitative structure-property and activity relationships are solely concerned with the forecasting of bio-activities and chemical or biological structure properties. Topological indices and physicochemical properties are used to aid prediction in this method. There is a significant body of literature related to the topological descriptors of chemical structure, which can be found in [17]; for example, the authors of [18] discussed some various types of nanotubes and nanostructures, the authors of [19, 20] studied some chemical networks like honeycomb networks and nanotubes, and different structures made by basic operations of linear algebras are discussed in [21]. For more studies on this topic, we suggest to refer [22–25]. For some relevant topics, we offer the literature found in [26–29]. In the following section of this paper, there are some basic formulas for predicting the physical properties of a network and system.

*Definition 1. *Hosoya polynomials, the famous most and very first, were introduced in 1988, and in 2015, a modified polynomial, also known as M-polynomial, is introduced by [30]. This form of the polynomial has a closed relationship to degree-based topological indices. Topological indices from M-polynomials of a graph may be obtained using a specific format. This M-polynomial may be defined as follows:where is considered as the size of graph given that .

*Definition 2. *(see [31, 32]). The first Zagreb index of graph is

*Definition 3. *(see [31, 32]). The second Zagreb index of graph is

*Definition 4. *(see [33]). The M-polynomial of the first and the second Zagreb indices is

*Definition 5. *(see [31–33]). The general Randić index and its M-polynomials are

*Definition 6. *(see [33]). The operators are defined as follows:The primary goal or participation of this study work is to investigate some vertex-degree-based M-polynomials of *Y*-shaped junctions and their variants, as well as some comprehensive applications of the theoretical and practical approach of *Y*-shaped junctions and their variants. We will investigate the structure of *Y*-shaped junctions and their variants in the following section. We will also present some key findings on the vertex-degree-based M-polynomials of *Y*-shaped junctions and their variants. Finally, we will reach a conclusion. There would be some interesting and important references to literature. To entice the reader to investigate Y-shaped junctions and their variants, vertex-degree-based M-polynomials, topological index are discussed in this research work. Furthermore, we also consider to discuss the applications for the Y-shaped junctions and their variants.

#### 2. *Y*-Junction Graphs and Methodology of Presented Work

For the aims of computation, the structures of the *Y*-junction are transformed in the vertex-edge graph, and for this, we follow the methodology presented in [34, 35]. The vertex-degree-based edge types are measured after getting the molecular graphs of junctions and their variants. These vertex-degree-based specifications of edges are necessary tools to compute our main results.

The *Y*-junctions investigated in this study are formed by the covalent connectivity of three armchair single-walled carbon nanotubes of finite length, and these nanotubes crossed at an angle of 120 degrees and are determined by a chiral vector .

Let for the parameter are integers and . A *Y*-shaped junction graph is assembled by making an armchair , which is called as branching point, and three carbon nanotubes of same length of -layers of hexagons. It has number of vertices, and the edges are in total. It has , degree three vertices, and only degree 2 vertices. Each nanotube comprises of faces of hexagons while branching point comprises of faces. In these faces, hexagons, six heptagons, and only three faces were nanotubes joined with branching point.

The notation is referred as graph of *Y*-shaped junction, and it contained no vertex of one degree. The notation is an extension of graph of *Y*-shaped junction as shown in Figure 2. This graph appears when one of the three nanotubes has exactly degree one vertices. The total number of vertices and edges of are and , accordingly. The graph with notation is another extension of junction graph. It contains edges and vertices. This graph is obtained by having degree one vertices on any two nanotubes of . In total, degree one vertices attaching to all three nanotubes of made another variant *Y*-junction, and it is known as . This newly developed graph contains edges and vertices.

The structure shown in Figure 1 is a front view of branching point with . There are three sides where armchair carbon nanotubes with different lengths, which is , will be attached. The structure shown in the Figure 2 is a variant of *Y*-junction having pendent or one degree vertices, and the notation used is . In the same figure, a place on a single nanotube is pointed where the extension of one degree , and vertices can be attached for different variants of *Y*-junctions, and these variants are denoted as , and , respectively. Without any one degree vertex, there will be original *Y*-junction of notation .

Some vertex-degree-based M-polynomials for *Y*-junction and its all variants are as follows.

Theorem 1. *Let be a structure of Y-type junction with and is the general M-polynomials for . Then,*

*Proof. *From Figure 2, which is the construction for the structure of *Y*-type junction, we can observe that there are three edge partitions based on the degree of end vertices of each edge that is defined as follows:The numbers of these edge types are , , and . Then, from the Definition 1, the M-polynomial of can be found as

Lemma 1. *Let be a structure of Y-type junction with , and are the differential operators for . Then,*

*Proof. *In this proof, we will provide an example of the usage of differential operators. Differentiating the equation given in the proof of Theorem 1 with respect to variable and times the result with the same variable. We will have the required result of the operator , for the structure of the *Y*-type junction which is .Similarly, by differentiating the equation given in the proof of Theorem 1 with respect to variable and times the result with the same variable, we will have the required result of the operator , for the structure of *Y*-type junction which is .

Lemma 2. *Let be a structure of Y-type junction with , and are the integral operators for . Then,*

*Proof. *In this proof, we will provide an example of the usage of integral operators. Introducing a new parameter in the place of a variable , and integrating the equation given in the proof of Theorem 1 with respect to the same variable, let say and times the result with the same variable. We will have the required result of the operator , for the structure of the *Y*-type junction which is .Similarly, by introducing a new parameter in the place of variable and integrating the equation given in the proof of Theorem 1 with respect to the same variable, let say and times the result with same variable, we will have the required result of operator , for the structure of *Y*-type junction which is .

Theorem 2. *Let be a structure of Y-type junction with and is the first Zagreb M-polynomials for . Then,*

*Proof. *A method to compute the first Zagreb M-polynomial, given in the equation (4) of Definition 4, and this methodology is derived from the basic formula of the first Zagreb index given in the equation (2) of Definition 2. Now, by using the differential operator of defined in the Lemma 1 and applying on the equation (4), we will have the required result of first Zagreb M-polynomial of , which is computed as follows:

Theorem 3. *Let be a structure of Y-type junction with , and is the second Zagreb M-polynomials for . Then,*

*Proof. *A method to compute the second Zagreb M-polynomial, given in the equation (5) of Definition 4, and this methodology is derived from the basic formula of the second Zagreb index given in the equation (3) of Definition 3. Now using the differential operator of , defined in the Lemma 1 and applying it on the equation (5), we will have the required result of the second Zagreb M-polynomial of , which is computed as follows:

Theorem 4. *Let be a structure of Y-type junction with , and is the general Randić M-polynomials for . Then,*

*Proof. *A method to compute the generalized Randić M-polynomial is given in equation (7) of Definition 5, and this methodology is derived from the basic formula of the generalized Randić index as given in equation (6) of Definition 5. Now, by using the generalized view of differential operators of , defined in the Lemma 1, and applying it on equation (7), we will have the required result of generalized Randić M-polynomial of , which is computed as follows:

#### 3. *I*^{′} Structure of *Y*-Second Type Junction

Theorem 5. *Let be a structure of Y-second type junction with and is the general M-polynomials for . Then,*

*Proof. *From Figure 2, which is the construction for the structure of *Y*-second type junction, we can observe that there are four edge partitions based on the degree of end vertices of each edge that is defined asThe numbers of these edge types are , and . Then from the Definition 1, the M-polynomial of can be found as

Lemma 3. *Let be a structure of Y-second type junction with , and are the differential operators for . Then,*

*Proof. *In this proof, we will provide an example for the usage of differential operators. By differentiating the equation given in the proof of Theorem 5 with respect to variable and times the result with same variable, we will have the required result of operator , for the structure of *Y*-type junction which is .Similarly, by differentiating the equation given in the proof of Theorem 5 with respect to variable and times the result with same variable, we will have the required result of operator , for the structure of *Y*-type junction which is .

Lemma 4. *Let be a structure of Y-second type junction with , and are the integral operators for . Then,*

*Proof. *In this proof, we will provide an example for the usage of integral operators. Introducing a new parameter in the place of variable , and integrating the equation given in the proof of Theorem 5 with respect to the same variable, let say and times the result with the same variable. We will have the required result of operator , for the structure of *Y*-type junction which is .Similarly, introducing a new parameter in the place of variable , and integrating the equation given in the proof of Theorem 5 with respect to the same variable, let say and times the result with the same variable. We will have the required result of operator , for the structure of the *Y*-type junction which is .

Theorem 6. *Let be a structure of Y-second type junction with and is the first Zagreb M-polynomials for . Then*

*Proof. *A method to compute the first Zagreb M-polynomial is given in equation (4) of Definition 4, and this methodology is derived from the basic formula of the first Zagreb index given as shown in equation (2) of Definition 2. Now, by using the differential operator of , defined in the Lemma 3, and applying it to equation (4), we will have the required result of the first Zagreb M-polynomial of , which is computed as given follows:

Theorem 7. *Let be a structure of Y-second type junction with and is the second Zagreb M-polynomials for . Then,*

*Proof. *A method to compute the second Zagreb M-polynomial is given in equation (5) of Definition 4, and this methodology is derived from the basic formula of the second Zagreb index given in equation (3) of Definition 3. Now, by using the differential operator of , defined in the Lemma 3 and applying it to equation (5), we will have the required result of the second Zagreb M-polynomial of , which is computed as follows:

Theorem 8. *Let be a structure of Y-second type junction with , and is the general Randić M-polynomials for . Then,*

*Proof. *A method to compute the generalized Randić M-polynomial is given in the following equation of Definition 5, and this methodology is derived from the basic formula of the generalized Randić index as given in equation (6) of Definition 5. Now, by using the generalized view of differential operators of , defined in the Lemma 3, and applying it to equation (7), we will have the required result of generalized Randić M-polynomial of , which is computed as follows:

#### 4. *I*^{″} Structure of *Y*-Third Type Junction

Theorem 9. *Let be a structure of Y-third type junction with and is the general M-polynomials for . Then,*

*Proof. *From Figure 2, which is the construction for the structure of the *Y*-second type junction, we can observe that there are four edge partitions based on the degree of end vertices of each edge that is defined as follows:The numbers of these edge types are , , and . Then, from the Definition 1, the M-polynomial of can be found as

Lemma 5. *Let be a structure of Y-third type junction with , and are the differential operators for . Then,*

*Proof. *In this proof, we will provide an example for the usage of differential operators. By differentiating the equation given in the proof of Theorem 9 with respect to variable and times the result with the same variable, we will have the required result of the operator , for the structure of *Y*-type junction, which is .Similarly, by differentiating the equation given in the proof of Theorem 9 with respect to variable and times the result with the same variable, we will have the required result of the operator , for the structure of *Y*-type junction which is .

Lemma 6. *Let be a structure of Y-third type junction with , and are the integral operators for . Then,*

*Proof. *In this proof, we will provide an example for the usage of integral operators. By introducing a new parameter in the place of a variable and integrating the equation given in the proof of Theorem 9 with respect to the same variable, let say and times the result with the same variable, we will have the required result of operator , for the structure of *Y*-type junction which is .Similarly, by introducing a new parameter in the place of variable and integrating the equation given in the proof of Theorem 9 with respect to the same variable, let say and times the result with same variable, we will have the required result of operator , for the structure of *Y*-type junction which is .