# Verification of Some Topological Indices ofY-Junction Based Nanostructures by M-Polynomials

Authors: Muhammad Azeem; Muhammad Kamran Jamil; Aisha Javed; Ali Ahmad
Journal: Journal of Mathematics
 (2022)

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## 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 ofY-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.


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## Body

## 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 asL, 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 discussingY-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.Figure 1
A branching pointY6,6.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:(1)Πχ;x,y=∑i≤jαi,jχxiyj,where αi,jχ is considered as the size of graph χ given that i≤j.Definition 2.
(see [31, 32]). The first Zagreb index of graph χ is(2)M1χ=∑uv∈Eχdu+dv.Definition 3.
(see [31, 32]). The second Zagreb index of graph χ is(3)M2χ=∑uv∈Eχdu⋅dv.Definition 4.
(see [33]). The M-polynomial of the first and the second Zagreb indices is(4)∏M1χ=Dx+DyΠχ;x,y,(5)∏M2χ=Dx+DyΠχ;x,y.Definition 5.
(see [31–33]). The general Randić index and its M-polynomials are(6)Rαχ=∑uv∈Eχdu⋅dvα,(7)∏Rαχ=DxαDyαΠχ;x,y.Definition 6.
(see [33]). The operators are defined as follows:(8)DxΠχ;x,y=x∂Πχ;x,y∂x,(9)DyΠχ;x,y=y∂Πχ;x,y∂y,(10)SxΠχ;x,y=∫0xΠχ;z,yzdz,(11)SyΠχ;x,y=∫0yΠχ;x,zzdz.
The primary goal or participation of this study work is to investigate some vertex-degree-based M-polynomials ofY-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 theY-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.TheY-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 n,n.Letk≥2 for the parameter n=2k are integers and m≥1. A Y-shaped junction graph Jmn,n is assembled by making an armchair Yn,n, which is called as branching point, and three carbon nanotubes Tmn,n of same length of m-layers of hexagons. It has 6k2+18k+24mk+6 number of vertices, and the edges are 9k2+21k+36mk+9 in total. It has 6k2+6k+24mk+6, degree three vertices, and only 12k degree 2 vertices. Each nanotube comprises of 4mk faces of hexagons while branching point comprises of 3k2−3k+5 faces. In these faces, 3k2−3k−4 hexagons, six heptagons, and only three faces were nanotubes joined with branching point.The notationJmn,n is referred as graph of Y-shaped junction, and it contained no vertex of one degree. The notation I′Jmn,n is an extension of graph of Y-shaped junction Jmn,n as shown in Figure 2. This graph I′Jmn,n appears when one of the three nanotubes has exactly 4k degree one vertices. The total number of vertices and edges of I′Jmn,n are 24mk+6k2+22k+6 and 36mk+9k2+25k+9, accordingly. The graph with notation I″Jmn,n is another extension of Jmn,n junction graph. It contains 36mk+9k2+29k+9 edges and 24mk+6k2+26k+6 vertices. This graph is obtained by having 8k degree one vertices on any two nanotubes of Jmn,n. In total, 12k degree one vertices attaching to all three nanotubes of Jmn,n made another variant Y-junction, and it is known as I‴Jmn,n. This newly developed graph contains 36mk+9k2+33k+9 edges and 24mk+6k2+30k+6 vertices.Figure 2
A variant ofY-junction I′Jmn,n.The structure shown in Figure1 is a front view of branching point Yn,n with n=6. There are three sides where armchair carbon nanotubes Tmn,n with different lengths, which is m≥1, will be attached. The structure shown in the Figure 2 is a variant of Y-junction having 4k pendent or one degree vertices, and the notation used is I′Jmn,n. In the same figure, a place on a single nanotube is pointed where the extension of one degree 4k,8k, and 12k vertices can be attached for different variants of Y-junctions, and these variants are denoted as I′Jmn,n,I″Jmn,n, and I‴Jmn,n, respectively. Without any one degree vertex, there will be original Y-junction of notation Jmn,n.Some vertex-degree-based M-polynomials forY-junction and its all variants are as follows.Theorem 1.
LetJmn,n be a structure of Y-type junction with k≥2,n=2k,m≥1 and ΠJmn,n;x,y is the general M-polynomials for Jmn,n. Then,(12)ΠJmn,n;x,y=6kx2y2+12kx2y3+9k2+3k+36mk+9x3y3.Proof.
From Figure2, 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:(13)e2,2=uv∈EJmn,n:du=dv=2,(14)e2,3=uv∈EJmn,n:du=2,dv=3,(15)e3,3=uv∈EJmn,n:du=dv=3.
The numbers of these edge types aree2,2=6k, e2,3=12k, and e3,3=9k2+3k+36mk+9. Then, from the Definition 1, the M-polynomial of Jmn,n can be found as(16)ΠJmn,n;x,y=∑i≤jmi,jJmn,nxiyj=6kx2y2+12kx2y3+9k2+3k+36mk+9x3y3.Lemma 1.
LetJmn,n be a structure of Y-type junction with k≥2,n=2k,m≥1, DxΠJmn,n;x,y and DyΠJmn,n;x,y are the differential operators for Jmn,n. Then,(17)DxΠJmn,n;x,y=12kx2y2+24kx2y3+27k2+9k+108mk+27x3y3,(18)DyΠJmn,n;x,y=12kx2y2+36kx2y3+27k2+9k+108mk+27x3y3.Proof.
In this proof, we will provide an example of the usage of differential operators. Differentiating the equation given in the proof of Theorem1 with respect to variable x and times the result with the same variable. We will have the required result of the operator DxΠJmn,n;x,y, for the structure of the Y-type junction which is Jmn,n.(19)DxΠJmn,n;x,y=x∂ΠJmn,n∂x=x∂∂x6kx2y2+12kx2y3+9k2+3k+36mk+9x3y3=x12kxy2+24kxy3+27k2+9k+108mk+27x2y3=12kx2y2+24kx2y3+27k2+9k+108mk+27x3y3.
Similarly, by differentiating the equation given in the proof of Theorem1 with respect to variable y and times the result with the same variable, we will have the required result of the operator DyΠJmn,n;x,y, for the structure of Y-type junction which is Jmn,n.(20)DyΠJmn,n;x,y=y∂ΠJmn,n∂y=y∂∂y6kx2y2+12kx2y3+9k2+3k+36mk+9x3y3=y12kx2y+36kx2y2+27k2+9k+108mk+27x3y2=12kx2y2+36kx2y3+27k2+9k+108mk+27x3y3.Lemma 2.
LetJmn,n be a structure of Y-type junction with k≥2,n=2k,m≥1, SxΠJmn,n;x,y and SyΠJmn,n;x,y are the integral operators for Jmn,n. Then,(21)SxΠJmn,n;x,y=3kx2y2+6kx2y3+3k2+k+18mk+3x3y3,(22)SyΠJmn,n;x,y=2kxy3+3kxy4+33k2+k+18mk+34xy4.Proof.
In this proof, we will provide an example of the usage of integral operators. Introducing a new parameter in the place of a variablex, and integrating the equation given in the proof of Theorem 1 with respect to the same variable, let say z and times the result with the same variable. We will have the required result of the operator SxΠJmn,n;x,y, for the structure of the Y-type junction which is Jmn,n.(23)SxΠJmn,n;x,y=∫0xΠJmn,n;z,yzdz=∫0x1z6kz2y2+12kz2y3+9k2+3k+36mk+9z3y3dz=∫0x6kzy2+∫0x12kzy3+∫0x9k2+3k+36mk+9z2y3dz=6kz22y2|0x+12kz22y3|0x+9k2+3k+36mk+9z33y3|0x=3kx2y2+6kx2y3+3k2+k+18mk+3x3y3.
Similarly, by introducing a new parameter in the place of variabley and integrating the equation given in the proof of Theorem 1 with respect to the same variable, let say z and times the result with same variable, we will have the required result of operator SyΠJmn,n;x,y, for the structure of Y-type junction which is Jmn,n.(24)SyΠJmn,n;x,y=∫0yΠJmn,n;x,zzdz=∫0y1z6kx2z2+12kx2z3+9k2+3k+36mk+9x3z3dz=∫0y6kxz2+∫0y12kxz3+∫0y9k2+3k+36mk+9x2z3dz=6kxz33|0y+12kxz44|0y+9k2+3k+36mk+9xz44|0y=2kxy3+3kxy4+33k2+k+18mk+34xy4.Theorem 2.
LetJmn,n be a structure of Y-type junction with k≥2,n=2k,m≥1 and ∏M1Jmn,n;x,y is the first Zagreb M-polynomials for Jmn,n. Then,(25)∏M1Jmn,n=24kx2y2+60k+x2y3+54k2+18k+216mk+54x3y3.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 Jmn,n defined in the Lemma 1 and applying on the equation (4), we will have the required result of first Zagreb M-polynomial of Jmn,n, which is computed as follows:(26)∏M1Jmn,n=Dx+DyΠJmn,n=DxΠJmn,n+DyΠJmn,n=12kx2y2+24kx2y3+27k2+9k+108mk+27x3y3+12kx2y2+36kx2y3+27k2+9k+108mk+27x3y3=24kx2y2+60kx2y3+54k2+18k+216mk+54x3y3.Theorem 3.
LetJmn,n be a structure of Y-type junction with k≥2,n=2k,m≥1, and ∏M2Jmn,n;x,y is the second Zagreb M-polynomials for Jmn,n. Then,(27)∏M2Jmn,n=24kx2y2+72kx2y3+327k2+9k+108mk+27x3y3.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 Jmn,n, 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 Jmn,n, which is computed as follows:(28)∏M2Jmn,n=DxDyΠJmn,n=Dx12kx2y2+36kx2y3+27k2+9k+108mk+27x3y3=24kx2y2+72kx2y3+327k2+9k+108mk+27x3y3.Theorem 4.
LetJmn,n be a structure of Y-type junction with k≥2,n=2k,m≥1, and ΠRαJmn,n;x,y is the general Randić M-polynomials for Jmn,n. Then,(29)∏RαJmn,n=4α6kx2y2+6α12kx2y3+9α9k2+3k+36mk+9x3y3.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 Jmn,n, defined in the Lemma 1, and applying it on equation (7), we will have the required result of generalized Randić M-polynomial of Jmn,n, which is computed as follows:(30)∏RαJmn,n=DxαDyαΠJmn,n=DxαDyα6kx2y2+12kx2y3+9k2+3k+36mk+9x3y3=Dxα2α6kx2y2+3α12kx2y3+3α9k2+3k+36mk+9x3y3=4α6kx2y2+6α12kx2y3+9α9k2+3k+36mk+9x3y3.


## 3.I′Jmn,n Structure of Y-Second Type Junction

Theorem 5.
LetI′Jmn,n be a structure of Y-second type junction with k≥2,n=2k,m≥1 and ∏I′Jmn,n;x,y is the general M-polynomials for I′Jmn,n. Then,(31)ΠI′Jmn,n;x,y=4kxy3+4kx2y2+8kx2y3+9k2+9k+36mk+9x3y3.Proof.
From Figure2, 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 as(32)e1,3=uv∈EI′Jmn,n:du=1,dv=3,e2,2=uv∈EI′Jmn,n:du=dv=2,(33)e2,3=uv∈EI′Jmn,n:du=2,dv=3,(34)e3,3=uv∈EI′Jmn,n:du=dv=3.
The numbers of these edge types aree1,3=4k,e2,2=4k, e2,3=8k and e3,3=9k2+9k+36mk+9. Then from the Definition 1, the M-polynomial of I′Jmn,n can be found as(35)∏I′Jmn,n;x,y=∑i≤jmi,jI′Jmn,nxiyj=4kxy3+4kx2y2+8kx2y3+9k2+9k+36mk+9x3y3.Lemma 3.
LetI′Jmn,n be a structure of Y-second type junction with k≥2,n=2k,m≥1, DxΠI′Jmn,n;x,y and DyΠI′Jmn,n;x,y are the differential operators for I′Jmn,n. Then,(36)DxΠI′Jmn,n;x,y=4kxy3+8kx2y2+16kx2y3+27k2+27k+108mk+27x3y3,(37)DyΠI′Jmn,n;x,y=12kxy3+8kx2y2+24kx2y3+27k2+27k+108mk+27x3y3.Proof.
In this proof, we will provide an example for the usage of differential operators. By differentiating the equation given in the proof of Theorem5 with respect to variable x and times the result with same variable, we will have the required result of operator DxΠI′Jmn,n;x,y, for the structure of Y-type junction which is I′Jmn,n.(38)DxΠI′Jmn,n;x,y=x∂ΠI′Jmn,n∂x=x∂∂x4kxy3+4kx2y2+8kx2y3+9k2+9k+36mk+9x3y3=x|4kxy3+8kx2y2+16kx2y3+27k2+27k+108mk+27x3y3=4kxy3+8kx2y2+16kx2y3+27k2+27k+108mk+27x3y3.
Similarly, by differentiating the equation given in the proof of Theorem5 with respect to variable y and times the result with same variable, we will have the required result of operator DyΠI′Jmn,n;x,y, for the structure of Y-type junction which is I′Jmn,n.(39)DyΠI′Jmn,n;x,y=y∂ΠI′Jmn,n∂y=y∂∂y4kxy3+4kx2y2+8kx2y3+9k2+9k+36mk+9x3y3=y12kxy3+8kx2y2+24kx2y3+27k2+27k+108mk+27x3y3=12kxy3+8kx2y2+24kx2y3+27k2+27k+108mk+27x3y3.Lemma 4.
LetI′Jmn,n be a structure of Y-second type junction with k≥2,n=2k,m≥1, SxΠI′Jmn,n;x,y and SyΠI′Jmn,n;x,y are the integral operators for I′Jmn,n. Then,(40)SxΠI′Jmn,n;x,y=4kxy3+2kx2y2+4kx2y3+3k2+3k+18mk+3x3y3,(41)SyΠI′Jmn,n;x,y=4k3xy3+2kx2y2+8k3x2y3+3k2+3k+18mk+3x3y3.Proof.
In this proof, we will provide an example for the usage of integral operators. Introducing a new parameter in the place of variablex, and integrating the equation given in the proof of Theorem 5 with respect to the same variable, let say z and times the result with the same variable. We will have the required result of operator SxΠI′Jmn,n;x,y, for the structure of Y-type junction which is I′Jmn,n.(42)SxΠI′Jmn,n;x,y=∫0xΠJmn,n;z,yzdz=∫0x1z4kzy3+4kz2y2+8kz2y3+9k2+9k+36mk+9z3y3dz=∫0x4ky3dz+∫0x4kzy2dz+∫0x8kzy3dz+∫0x9k2+9k+36mk+9z2y3dz=4kzy3|0x+4kz2y2|0x+8kz2y3|0x+9k2|0x+9k+36mk+9z3y3|0x6kz22y2|0x+12kz22y3|0x+9k2+3k+36mk+9z33y3|0x=4kxy3+2kx2y2+4kx2y3+3k2+3k+18mk+3x3y3.
Similarly, introducing a new parameter in the place of variabley, and integrating the equation given in the proof of Theorem 5 with respect to the same variable, let say z and times the result with the same variable. We will have the required result of operator SyΠI′Jmn,n;x,y, for the structure of the Y-type junction which is I′Jmn,n.(43)SyΠI′Jmn,n;x,y=∫0yΠJmn,n;x,zzdz=∫0y1z4kxz3+4kx2z2+8kx2z3+9k2+9k+36mk+9x3z3dz=∫0y4kxz2dz+∫0y4kx2zdz+∫0y8kx2z2dz+∫0y9k2+9k+36mk+9x3z2dz=4kxz3|0y+4kx2z2|0y+8kx2z3|0y+9k2|0y+9k+36mk+9x3z3|0y=4k3xy3+2kx2y2+8k3x2y3+3k2+3k+18mk+3x3y3.Theorem 6.
LetJmn,n be a structure of Y-second type junction with k≥2,n=2k,m≥1 and ΠM1I′Jmn,n;x,y is the first Zagreb M-polynomials for I′Jmn,n. Then(44)∏M1I′Jmn,n=16kxy3+16kx2y2+40kx2y3+54k2+54k+216mk+54x3y3.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 I′Jmn,n, defined in the Lemma 3, and applying it to equation (4), we will have the required result of the first Zagreb M-polynomial of I′Jmn,n, which is computed as given follows:(45)∏M1I′Jmn,n=Dx+DyΠI′Jmn,n=DxΠI′Jmn,n+DyΠI′Jmn,n=4kxy3+8kx2y2+16kx2y3+27k2+27k+108mk+27x3y3+12kxy3+8kx2y2+24kx2y3+27k2+27k+108mk+27x3y3=16kxy3+16kx2y2+40kx2y3+54k2+54k+216mk+54x3y3.Theorem 7.
LetJmn,n be a structure of Y-second type junction with k≥2,n=2k,m≥1 and ΠM2I′Jmn,n;x,y is the second Zagreb M-polynomials for I′Jmn,n. Then,(46)∏M2I′Jmn,n=12kxy3+16kx2y2+48kx2y3+81k2+81k+324mk+81x3y3.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 I′Jmn,n, defined in the Lemma 3 and applying it to equation (5), we will have the required result of the second Zagreb M-polynomial of I′Jmn,n, which is computed as follows:(47)∏M2I′Jmn,n=DxDyΠI′Jmn,n=Dx12kxy3+8kx2y2+24kx2y3+27k2+27k+108mk+27x3y3=12kxy3+16kx2y2+48kx2y3+81k2+81k+324mk+81x3y3.Theorem 8.
LetJmn,n be a structure of Y-second type junction with k≥2,n=2k,m≥1, and ΠRαI′Jmn,n;x,y is the general Randić M-polynomials for I′Jmn,n. Then,(48)∏RαI′Jmn,n=3α4kxy3+22α4kx2y2+6α8kx2y3+32α9k2+9k+36mk+9x3y3.Proof.
A method to compute the generalized Randić M-polynomial is given in the following equation of Definition5, 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 I′Jmn,n, defined in the Lemma 3, and applying it to equation (7), we will have the required result of generalized Randić M-polynomial of I′Jmn,n, which is computed as follows:(49)∏RαI′Jmn,n=DxαDyαΠI′Jmn,n=DxαDyα4kxy3+4kx2y2+8kx2y3+9k2+9k+36mk+9x3y3=3α4kxy3+22α4kx2y2+6α8kx2y3+32α9k2+9k+36mk+9x3y3.


## 4.I″Jmn,n Structure of Y-Third Type Junction

Theorem 9.
LetI″Jmn,n be a structure of Y-third type junction with k≥2,n=2k,m≥1 and ΠI″Jmn,n;x,y is the general M-polynomials for I″Jmn,n. Then,(50)ΠI″Jmn,n;x,y=8kxy3+2kx2y2+4kx2y3+9k2+15k+36mk+9x3y3.Proof.
From Figure2, 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:(51)e1,3=uv∈EI″Jmn,n:du=1,dv=3,e2,2=uv∈EI″Jmn,n:du=dv=2,(52)e2,3=uv∈EI″Jmn,n:du=2,dv=3,(53)e3,3=uv∈EI″Jmn,n:du=dv=3.
The numbers of these edge types aree1,3=8k,e2,2=2k, e2,3=4k, and e3,3=9k2+15k+36mk+9. Then, from the Definition 1, the M-polynomial of I″Jmn,n can be found as(54)ΠI″Jmn,n;x,y=∑i≤jmi,jI″Jmn,nxiyj=8kxy3+2kx2y2+4kx2y3+9k2+15k+36mk+9x3y3.Lemma 5.
LetI″Jmn,n be a structure of Y-third type junction with k≥2,n=2k,m≥1, DxΠI″Jmn,n;x,y and DyI″ΠJmn,n;x,y are the differential operators for I″Jmn,n. Then,(55)DxΠI″Jmn,n;x,y=8kxy3+4kx2y2+8kx2y3+27k2+45k+108mk+27x3y3,(56)DyΠI″Jmn,n;x,y=24kxy3+4kx2y2+12kx2y3+27k2+45k+108mk+27x3y3.Proof.
In this proof, we will provide an example for the usage of differential operators. By differentiating the equation given in the proof of Theorem9 with respect to variable x and times the result with the same variable, we will have the required result of the operator DxΠI″Jmn,n;x,y, for the structure of Y-type junction, which is I″Jmn,n.(57)DxΠI″Jmn,n;x,y=x∂ΠI″Jmn,n∂x=x∂∂x8kxy3+2kx2y2+4kx2y3+9k2+15k+36mk+9x3y3=x8ky3+4kxy2+8kxy3+27k2+45k+108mk+27x2y3=8kxy3+4kx2y2+8kx2y3+27k2+45k+108mk+27x3y3.
Similarly, by differentiating the equation given in the proof of Theorem9 with respect to variable y and times the result with the same variable, we will have the required result of the operator DyΠI″Jmn,n;x,y, for the structure of Y-type junction which is I″Jmn,n.(58)DyΠI″Jmn,n;x,y=y∂ΠI″Jmn,n∂y=y∂∂y8kxy3+2kx2y2+4kx2y3+9k2+15k+36mk+9x3y3=y24kxy2+4kx2y+12kx2y2+27k2+45k+108mk+27x3y2=24kxy3+4kx2y2+12kx2y3+27k2+45k+108mk+27x3y3.Lemma 6.
LetI″Jmn,n be a structure of Y-third type junction with k≥2,n=2k,m≥1, SxΠI″Jmn,n;x,y and SyI″ΠJmn,n;x,y are the integral operators for I″Jmn,n. Then,(59)SxΠI″Jmn,n;x,y=3kx2y2+6kx2y3+3k2+k+18mk+3x3y3,(60)SyΠI″Jmn,n;x,y=3kx2y2+4kx2y3+3k2+k+18mk+3x3y3.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 variablex and integrating the equation given in the proof of Theorem 9 with respect to the same variable, let say z and times the result with the same variable, we will have the required result of operator SxΠI″Jmn,n;x,y, for the structure of Y-type junction which is I″Jmn,n.(61)SxΠI″Jmn,n;x,y=∫0xΠI″Jmn,n;z,yzdz=∫0x1z8kzy3+2kz2y2+4kz2y3+9k2+15k+36mk+9z3y3dz=∫0x8ky3+∫0x2kzy2+∫0x4kzy3+∫0x9k2+15k+36mk+9z2y3dz=8kzy3|0x|+2kz2y2|0x+4kz2y3|0x+9k2+15k+36mk+9z3y3|0x=3kx2y2+6kx2y3+3k2+k+18mk+3x3y3.
Similarly, by introducing a new parameter in the place of variabley and integrating the equation given in the proof of Theorem 9 with respect to the same variable, let say z and times the result with same variable, we will have the required result of operator SyΠI″Jmn,n;x,y, for the structure of Y-type junction which is I″Jmn,n.(62)SyΠI″Jmn,n;x,y=∫0yΠJmn,n;x,zzdz=∫0y1z8kxz3+2kx2z2+4kx2z3+9k2+15k+36mk+9x3z3dz=∫0y8kxz2+∫0y2kx2z+∫0y4kx2z2+∫0y9k2+15k+36mk+9x3z2dz=8kxz2+∫0y2kx2z+∫0y4kx2z2+∫0y9k2+15k+36mk+9x3z2|0y=3kx2y2+4kx2y3+3k2+k+18mk+3x3y3.Theorem 10.
LetI″Jmn,n be a structure of Y-third type junction with k≥2,n=2k,m≥1 and ΠM1I″Jmn,n;x,y is the first Zagreb M-polynomials for I″Jmn,n. Then,(63)∏M1I″Jmn,n=32kxy3+8kx2y2+20kx2y3+54k2+90k+216mk+54x3y3.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 in equation (2) of Definition 2. Now, by using the differential operator of I″Jmn,n, defined in the Lemma 5 and applying it to the equation (4), we will have the required result of the first Zagreb M-polynomial of I″Jmn,n, which is computed as follows:(64)∏M1I″Jmn,n=Dx+DyΠI″Jmn,n=DxΠI″Jmn,n+DyΠI″Jmn,n=8kxy3+4kx2y2+8kx2y3+27k2+45k+108mk+27x3y3+24kxy3+4kx2y2+12kx2y3+27k2+45k+108mk+27x3y3=32kxy3+8kx2y2+20kx2y3+54k2+90k+216mk+54x3y3.Theorem 11.
LetI″Jmn,n be a structure of Y-third type junction with k≥2,n=2k,m≥1 and ΠM2I″Jmn,n;x,y is the second Zagreb M-polynomials for I″Jmn,n. Then,(65)∏M2I″Jmn,n=24kxy3+8kx2y2+24kx2y3+81k2+135k+324mk+81x3y3.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 I″Jmn,n, defined in the Lemma 5 and applying it to equation (5), we will have the required result of the second Zagreb M-polynomial of I″Jmn,n, which is computed as follows:(66)∏M2I″Jmn,n=DxDyΠI″Jmn,n=Dx24kxy3+4kx2y2+12kx2y3+27k2+45k+108mk+27x3y3=24kxy3+8kx2y2+24kx2y3+81k2+135k+324mk+81x3y3.Theorem 12.
LetI″Jmn,n be a structure of Y-third type junction with k≥2,n=2k,m≥1 and ΠRαI″Jmn,n;x,y is the general Randić M-polynomials for I″Jmn,n. Then,(67)∏RαI″Jmn,n=3α8kxy3+4α2kx2y2+6α4kx2y3+9α9k2+15k+36mk+9x3y3.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 given in equation (6) of Definition 5. Now, by using the generalized view of differential operators of I″Jmn,n, defined in the Lemma 5 and applying it to equation (7), we will have the required result of generalized Randić M-polynomial of I″Jmn,n, which is computed as follows:(68)∏RαI″Jmn,n=DxαDyαΠI″Jmn,n=DxαDyα8kxy3+2kx2y2+4kx2y3+9k2+15k+36mk+9x3y3=3α8kxy3+4α2kx2y2+6α4kx2y3+9α9k2+15k+36mk+9x3y3.


## 5.I‴Jmn,n Structure of Y-Fourth Type Junction

Theorem 13.
LetI‴Jmn,n be a structure of Y-fourth type junction with k≥2,n=2k,m≥1 and ΠI‴Jmn,n;x,y is the general M-polynomials for I‴Jmn,n. Then,(69)ΠI‴Jmn,n;x,y=12kxy3+9k2+21k+36mk+9x3y3.Proof.
From Figure2, which is the construction for the structure of the Y-second type junction, we can observe that there are two edge partitions based on the degree of end vertices of each edge that is defined as(70)e1,3=uv∈EI‴Jmn,n:du=1,dv=3,e3,3=uv∈EI‴Jmn,n:du=dv=3.
The numbers of these edge types aree1,3=12k and e3,3=9k2+21k+36mk+9. Then, from the Definition 1, the M-polynomial of I‴Jmn,n can be found as(71)ΠI″″Jmn,n;x,y=∑i≤jmi,jI″″Jmn,nxiyj=12kxy3+9k2+21k+36mk+9x3y3.Lemma 7.
LetI‴Jmn,n be a structure of Y-fourth type junction with k≥2,n=2k,m≥1, DxΠI‴Jmn,n;x,y and DyI‴ΠJmn,n;x,y are the integral operators for I‴Jmn,n. Then,(72)DxΠI‴Jmn,n;x,y=12kxy3+27k2+63k+108mk+27x3y3,(73)DyΠI‴Jmn,n;x,y=36kxy3+27k2+63k+108mk+27x3y3.Proof.
In this proof, we will provide an example for the usage of differential operators. By differentiating the equation given in the proof of Theorem13 with respect to variable x and times the result with the same variable, we will have the required result of the operator DxΠI‴Jmn,n;x,y, for the structure of Y-type junction which is I‴Jmn,n.(74)DxΠI‴Jmn,n;x,y=x∂ΠI‴Jmn,n∂x=x∂∂x12kxy3+9k2+21k+36mk+9x3y3=x12ky3+27k2+63k+108mk+27x2y3=12kxy3+27k2+63k+108mk+27x3y3.
Similarly, by differentiating the equation given in the proof of Theorem13 with respect to variable y and times the result with the same variable, we will have the required result of the operator DyΠI‴Jmn,n;x,y, for the structure of Y-type junction which is I‴Jmn,n.(75)DyΠI‴Jmn,n;x,y=y∂ΠI‴Jmn,n∂y=y∂∂x12kxy3+9k2+21k+36mk+9x3y2=y36kxy2+27k2+63k+108mk+27x3y2=36kxy3+27k2+63k+108mk+27x3y3.Lemma 8.
LetI‴Jmn,n be a structure of Y-fourth type junction with k≥2,n=2k,m≥1, SxΠI‴Jmn,n;x,y and SyI‴ΠJmn,n;x,y are the integral operators for I‴Jmn,n. Then,(76)SxΠI‴Jmn,n;x,y=3kx2y2+6kx2y3+3k2+k+18mk+3x3y3,(77)Sy=3kx2y2+4kx2y3+3k2+k+18mk+3x3y3.Proof.
In this proof, we will provide an example for the usage of integral operators. By introducing a new parameter in the place of variablex and integrating the equation given in the proof of Theorem 13 with respect to the same variable, let say z and times the result with the same variable, we will have the required result of the operator SxΠI‴Jmn,n;x,y, for the structure of Y-type junction which is I′′′Jmn,n.(78)SxΠI‴Jmn,n;x,y=∫0xΠI‴Jmn,n;z,yzdz=∫0x1z12kzy3+9k2+21k+36mk+9z3y3dz=∫0x12ky3+∫0x9k2+21k+36mk+9z2y3z=12kzy3|0x+9k2+21k+36mk+9z3y3|0x=3kx2y2+6kx2y3+3k2+k+18mk+3x3y3.
Similarly, by introducing a new parameter in the place of variabley, and integrating the equation given in the proof of Theorem 13 with respect to the same variable, let say z and times the result with the same variable, we will have the required result of the operator SyΠI‴Jmn,n;x,y, for the structure of Y-type junction which is I‴Jmn,n.(79)SyΠI‴Jmn,n;x,y=∫0xΠI‴Jmn,n;x,zzdz=∫0y1z12kxz3+9k2+21k+36mk+9x3z3dz=∫0y12kxz2dz+∫0y9k2+21k+36mk+9x3z2zdz=12kxz33|0y+9k2+21k+36mk+9x3z33|0y=3kx2y2+4kx2y3+3k2+k+18mk+3x3y3.Theorem 14.
LetI‴Jmn,n be a structure of Y-fourth type junction with k≥2,n=2k,m≥1 and ΠM1I‴Jmn,n;x,y is the first Zagreb M-polynomials for I‴Jmn,n. Then,(80)∏M1I‴Jmn,n=48kxy3+54k2+126k+216mk+54x3y3.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 in equation (2) of Definition 2. Now, by using the differential operator of I‴Jmn,n, defined in the Lemma 7 and applying it to Equation 4, we will have the required result of the first Zagreb M-polynomial of I‴Jmn,n, which is computed as follows:(81)∏M1I‴Jmn,n=Dx+DyΠI‴Jmn,n=DxΠI‴Jmn,n+DyΠI‴Jmn,n=12kxy3+27k2+63k+108mk+27x3y3+36kxy3+27k2+63k+108mk+27x3y3=48kxy3+54k2+126k+216mk+54x3y3.Theorem 15.
LetI‴Jmn,n be a structure of Y-fourth type junction with k≥2,n=2k,m≥1 and ΠM2I‴Jmn,n;x,y is the second Zagreb M-polynomials for I‴Jmn,n. Then,(82)∏M2I‴Jmn,n=36kxy3+81k2+189k+324mk+27x3y3.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 I‴Jmn,n, defined in the Lemma 7 and applying it to equation (5), we will have the required result of the second Zagreb M-polynomial of I‴Jmn,n, which is computed as follows:(83)∏M2I‴Jmn,n=DxDyΠI‴Jmn,n=Dx36kxy3+27k2+63k+108mk+9x3y3=36kxy3+81k2+189k+324mk+27x3y3.Theorem 16.
LetI‴Jmn,n be a structure of Y-fourth type junction with k≥2,n=2k,m≥1 and ΠRαI‴Jmn,n;x,y is the general Randić M-polynomials for I‴Jmn,n. Then,(84)∏RαI‴Jmn,n=3α12kxy3+32α9k2+21k+36mk+9x3y3.Proof.
A method to compute the generalized Randić M-polynomial is given in equation (7) of Definition 1, and this methodology is derived from the basic formula of the generalized Randić index given in equation (6) of Definition 5. Now, by using the generalized view of differential operators of I‴Jmn,n, defined in the Lemma 7, and applying it to equation (7), we will have the required result of generalized Randić M-polynomial of I‴Jmn,n, which is computed as follows:(85)∏RαI‴Jmn,n=DxαDyαΠI‴Jmn,n=DxαDyα12kxy3+9k2+21k+36mk+9x3y3=3α12kxy3+32α9k2+21k+36mk+9x3y3.


## 6. Verifications of Topological Indices

There are some techniques to find the authenticity or correctness of topological indices. M-polynomial is one of the techniques. By using Theorems2 to 4, we can verify the main results of topological indices given in the [36], for the structure of Y-type junction or Jmn,n. Similarly, for the results of the structure of Y-second type junction or I′Jmn,n, we can use the results of our main Theorems 6 to 8 and use the values of x=y=1. For the results of the structure of the Y-third type junction or I″Jmn,n, we can use the results of our main Theorems 10 to 12 while Theorems 14 to 16 are used to verify the results of the structure of the Y-fourth type junction or I‴Jmn,n, given as follows.Theorem 17.
LetJmn,n be a structure of Y-type junction with k≥2,n=2k,m≥1 and M1Jmn,n is the first Zagreb index for Jmn,n. Then,(86)M1Jmn,n=102k+54k2+216mk+54.Theorem 18.
LetJmn,n be a structure of Y-type junction with k≥2,n=2k,m≥1 and M2Jmn,n is the second Zagreb index for Jmn,n. Then,(87)M2Jmn,n=123k+81k2+324mk+81.Theorem 19.
LetJmn,n be a structure of Y-type junction with k≥2,n=2k,m≥1 and RαJmn,n is the general Randić index for Jmn,n. Then,(88)RαJmn,n=4α6k+6α12k+9α9k2+3k+36mk+9.Theorem 20.
LetJmn,n be a structure of Y-second type junction with k≥2,n=2k,m≥1 and M1I′Jmn,n is the first Zagreb index for I′Jmn,n. Then,(89)M1I′Jmn,n=126k+54k2+216mk+54.Theorem 21.
LetJmn,n be a structure of Y-second type junction with k≥2,n=2k,m≥1 and M2I′Jmn,n is the second Zagreb index for I′Jmn,n. Then,(90)M2I′Jmn,n=157k+81k2+324mk+81.Theorem 22.
LetJmn,n be a structure of Y-second type junction with k≥2,n=2k,m≥1 and RαI′Jmn,n is the general Randić index for I′Jmn,n. Then,(91)RαI′Jmn,n=3α4k+22α4k+6α8k+32α9k2+9k+36mk+9.Theorem 23.
LetI′′Jmn,n be a structure of Y-third type junction with k≥2,n=2k,m≥1 and M1I″Jmn,n is the first Zagreb index for I″Jmn,n. Then,(92)M1I″Jmn,n=150k+54k2+216mk+54.Theorem 24.
LetI″Jmn,n be a structure of Y-third type junction with k≥2,n=2k,m≥1 and M2I″Jmn,n is the second Zagreb index for I″Jmn,n. Then,(93)M2I″Jmn,n=191k+81k2+324mk+81.Theorem 25.
LetI″Jmn,n be a structure of Y-third type junction with k≥2,n=2k,m≥1 and RαI″Jmn,n is the general Randić index for I″Jmn,n. Then,(94)RαI″Jmn,n=3α8k+4α2k+6α4k+9α9k2+15k+36mk+9.Theorem 26.
LetI‴Jmn,n be a structure of Y-fourth type junction with k≥2,n=2k,m≥1 and M1I‴Jmn,n is the first Zagreb index for I‴Jmn,n. Then,(95)M1I‴Jmn,n=174k+54k2+216mk+54.Theorem 27.
LetI‴Jmn,n be a structure of Y-fourth type junction with k≥2,n=2k,m≥1 and M2I‴Jmn,n is the second Zagreb index for I‴Jmn,n. Then,(96)M2I‴Jmn,n=225k+81k2+324mk+81.Theorem 28.
LetI‴Jmn,n be a structure of Y-fourth type junction with k≥2,n=2k,m≥1 and RαI‴Jmn,n is the general Randić index for I‴Jmn,n. Then,(97)RαI‴Jmn,n=3α12k+32α9k2+15k+36mk+9.
From the above discussion, the topological indices, given in the [36], are found correct, and calculations are verified here with M-polynomials methodology.


## 7. Conclusion

In this research work, vertex-degree-based M-polynomials ofY-junctions and their variants are studied for the first time. We determined general Randić first and second Zagreb vertex-degree-based M-polynomials for four types of Y-shaped carbon nanotube junctions Jmn,n. By this method, Y-junctions and their structures are elaborated in numerical form, and the whole compound is described as a numeric digit. Instead of a whole complex structure, it will be easy to see as a numeric quantity. Furthermore, we verified the results from literature given on the concept of the topological index from the method of vertex-degree-based M-polynomials and concluded that the results are calculated correctly in the literature.



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*Source: 8238651-2022-07-04.xml*