# Comparative Study ofY-Junction Nanotubes with Vertex-Edge Based Topological Descriptors

Authors: Al-Nashri Al-Hossain Ahmad
Journal: Journal of Mathematics
 (2022)

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## Abstract

The current results of various forms of carbon nanostructures and its applications in different areas attract the researchers. In pharmaceutical, medicine, industry and electronic devices they used it by its graphical invariants. The detection of different types of carbon nanotubes junctions enhanced the attention and interest for forthcoming devices like transistors and amplifiers. A topological index plays a very important role in the study of physicochemical properties of biological and chemical structures. In this paper, we determine results ofve-degree topological indices for various type of carbon nanotubes Y-junctions and their comparisons. The particular indices called as The first ve-degree Zagreb β index, the second ve-degree Zagreb index, ve-degree Randic index, ve-degree atom-bond connectivity index, ve-degree geometric-arithmetic index, ve-degree harmonic index and ve-degree sum-connectivity index.


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## Body

## 1. Introduction

Let a graph having vertex setV and edge set E possesses the properties of connectivity, usually labeled as G=V,E. For a vertex x1∈V, the concept of open neighborhood of that vertex x1 is formulated as Nx1=x2∈V:x1x2∈E, while the concept of closed neighborhood formulated and notated by Nx1=Nx1∪x1, [1–3]. A notation ξvex1, is used for the ve-degree of any vertex x1∈V, and measured by the count of distinct edges which are incident to any vertex from the closed neighborhood of x1. Further detail and discussion on this notation and its mathematical definition, one can see [4–6].In molecular graph theory, vertices and edges are replaced by atoms and their bonds while transforming from a molecular structure to a molecular graph, respectively, [7, 8]. Carbon nanotubes with branching ends are promising building blocks for next-generation enhanced nanoelectronics and nanodevices. In the junction family, three-terminal devices and carbon nanotube graphs have tremendous potential. While study the chemical things for various determinations in different areas, the energy bond is the one of the most important thermophysical to be measured. There are different type of nanotubes junctions for example, X, Y, L and T and their applications can be seen in [9–12].The topological descriptor of a given graph is a numeric number that describes the quantitative structural-property relationship and quantitative structural-activity of the molecular graph [13–16]. The researcher in [17] discussed the metal-organic network, supramolecular chain is discussed by [18], carbon nanotubes are measured in [19] with different parameters of graph-based chemical theory. For study of different types of topological indices, see [20–25]. Some new variants and generalized results on the topological descriptors are found in the articles suggested [3, 26, 27].There are variety of topological descriptors, one of them is the vertex-edge based that will be discussed in this article. The researchers in [1], defined the “ev-degree,” and [4] contributed in this study. Basic definitions regarding “ve-degree” topological indices, refer to [28].The vertex-edge based topological descriptors are: The firstve-degree Zagreb β index Mβve1Ymn,n=∑x1x2∈Eξvex1+ξvex2, the second ve-degree Zagreb index Mve2Ymn,n=∑x1x2∈Eξvex1×ξvex2, ve-degree Randić index RveYmn,n=∑x1x2∈Eξvex1×ξvex2−1/2, ve-degree atom-bond connectivity index ABCveYmn,n=∑x1x2∈Eξvex1+ξvex2−2/ξvex1×ξvex21/2, ve-degree geometric-arithmetic index GAveYmn,n=∑x1x2∈E2ξvex1×ξvex21/2/ξvex1+ξvex2, ve-degree harmonic index HveYmn,n=∑x1x2∈E2/ξvex1+ξvex2 and ve-degree sum-connectivity index χveYmn,n=∑x1x2∈Eξvex1+ξvex2−1/2. For further results and detail on vertex-edge based topological indices see [29, 30]. Some other related topics based on the information of edges of a graph are detailed in [31–37]. In this paper, the exact values of vertex-edge based topological indices for Y-junctions carbon nanotubes are determined.


## 2.Y-Junction Graphs

TheY-junctions investigated in this paper are formed by the covalent interconnection of three finite-length armchair single-walled nanotubes that intersect at a 120° angle and are specified by the chiral vector n,n. For detail study of structures, authors refer to [38–41]. A Y-junction graph is defined as follows:Letn is even and m,n are two integers. A graph of Y-junction labeled as Ymn,n is constructed by using an armchair Yn,n, and three CNTsTmn,n single-walled armchair which are identical having m hexagons-layers. Total face count is 3n2/4−3n/2+5 in an armchair Yn,n, containing openings of count three, heptagons count is six, hexagons count is 3n2/4−3n/2−5. Furthermore, each armchair tube labeled by Tmn,n contained hexagonal-faces of count 2mn. The degree two count vertices are 6n, degree three with count 3n2/2+3n+12mn+6, collectively having 3n2/2+9n+12mn+6 order, and 9n2/4+21n/2+18mn+9 size.In this work, a junction graph labeled withYmn,n is graphs having no pendent or degree one vertex, exists. This work also consists of other three topologies of Y-junction graphs labeled with Ym1n,n,Ym2n,n and Ym3n,n and these contained some vertices with degree one. These further topologies are constructed by Ymn,n-junction graphs by adding pendants to degree 2 vertices. Single tube among three tubes of Ymn,n has exactly 2n count of vertices having two degree. In result, 6n is the maximum number of pendants that can be utilised with this attachment for Ymn,n and 2n for each tube.


## 3. Theve-Degree Results of Y-Junction Graph Ymn,n

This section presented theve-degree results of Y-junction graph Ymn,n. This graph does not contain any pendent vertex that is shown in Figure 1. The edge partition of end vertices ve-degree of each edge along with the degree of end vertices of each edge for Ymn,n graph is given in Table 1.Figure 1
A variant ofY-junction Ymn,n.Table 1
Theve-degrees of each edge of Ymn,n junction graph.
ξx1,ξx2ξvex1,ξvex2Count2,25,53n2,35,86n3,38,96n3,39,99n2/4−9n/2+18mn+9


### 3.1. The Firstve-Degree Zagreb β Index

(1)Mβve1Ymn,n=∑x1x2∈Eξvex1+ξvex2=103n+136n+176n+189n24−9n2+18mn+9=81n22+129n+324mn+162.


### 3.2. The Secondve-Degree Zagreb Index

(2)Mve2Ymn,n=∑x1x2∈Eξvex1×ξvex2=253n+406n+726n+819n24−9n2+18mn+9=729n22+765n2+1458mn+729.


### 3.3. The Randić Index Developed byve-Degree Methodology

Utilizing the edge-partition details described in the Table1, we measured the Randić index developed by ve-degree methodology:(3)RveYmn,n=∑x1x2∈Eξvex1×ξvex2−1/2=25−1/23n+40−1/26n+72−1/26n+81−1/29n24−9n2+18mn+9=n24+110+31010+22+2mn+1.


### 3.4. The Atom-Bond Connectivity Index Developed byve-Degree Methodology

Utilizing the edge-partition details described in the Table1, we measured the atom-bond connectivity index developed by ve-degree methodology:(4)ABCveYmn,n=∑x1x2∈Eξvex1+ξvex2−2ξvex1×ξvex2−1/2=3n825+6n1140+6n1572+9n24−9n2+18mn+91681=n2+625+311010+302+8m−2n+4.


### 3.5. The Geometric-Arithmetic Index Developed byve-Degree Methodology

Utilizing the edge-partition details described in the Table1, we measured the geometric-arithmetic index developed by ve-degree methodology:(5)GAveYmn,n=∑x1x2∈E2ξvex1×ξvex2−1/2ξvex1+ξvex2=3n22510+6n24013+6n27217+9n24−9n2+18mn+928118=9n24+−32+241013+72217+18mn+9.


### 3.6. The Harmonic Index Developed byve-Degree Methodology

Utilizing the edge-partition details described in the Table1, we measured the harmonic index developed by ve-degree methodology:(6)HveYmn,n=∑x1x2∈E2ξvex1+ξvex2=3n210+6n213+6n217+9n24−9n2+18mn+9218=n24+38212210+2mn+1.


### 3.7. The Sum-Connectivity Index Developed byve-Degree Methodology

Utilizing the edge-partition details described in the Table1, we measured the sum-connectivity index developed by ve-degree methodology:(7)χveYmn,n=∑x1x2∈Eξvex1+ξvex2−1/2=3n110+6n113+6n117+9n24−9n2+18mn+9118=32n28+31010+61313+61717+26−92+18mn+322.


## 3.1. The Firstve-Degree Zagreb β Index

(1)Mβve1Ymn,n=∑x1x2∈Eξvex1+ξvex2=103n+136n+176n+189n24−9n2+18mn+9=81n22+129n+324mn+162.


## 3.2. The Secondve-Degree Zagreb Index

(2)Mve2Ymn,n=∑x1x2∈Eξvex1×ξvex2=253n+406n+726n+819n24−9n2+18mn+9=729n22+765n2+1458mn+729.


## 3.3. The Randić Index Developed byve-Degree Methodology

Utilizing the edge-partition details described in the Table1, we measured the Randić index developed by ve-degree methodology:(3)RveYmn,n=∑x1x2∈Eξvex1×ξvex2−1/2=25−1/23n+40−1/26n+72−1/26n+81−1/29n24−9n2+18mn+9=n24+110+31010+22+2mn+1.


## 3.4. The Atom-Bond Connectivity Index Developed byve-Degree Methodology

Utilizing the edge-partition details described in the Table1, we measured the atom-bond connectivity index developed by ve-degree methodology:(4)ABCveYmn,n=∑x1x2∈Eξvex1+ξvex2−2ξvex1×ξvex2−1/2=3n825+6n1140+6n1572+9n24−9n2+18mn+91681=n2+625+311010+302+8m−2n+4.


## 3.5. The Geometric-Arithmetic Index Developed byve-Degree Methodology

Utilizing the edge-partition details described in the Table1, we measured the geometric-arithmetic index developed by ve-degree methodology:(5)GAveYmn,n=∑x1x2∈E2ξvex1×ξvex2−1/2ξvex1+ξvex2=3n22510+6n24013+6n27217+9n24−9n2+18mn+928118=9n24+−32+241013+72217+18mn+9.


## 3.6. The Harmonic Index Developed byve-Degree Methodology

Utilizing the edge-partition details described in the Table1, we measured the harmonic index developed by ve-degree methodology:(6)HveYmn,n=∑x1x2∈E2ξvex1+ξvex2=3n210+6n213+6n217+9n24−9n2+18mn+9218=n24+38212210+2mn+1.


## 3.7. The Sum-Connectivity Index Developed byve-Degree Methodology

Utilizing the edge-partition details described in the Table1, we measured the sum-connectivity index developed by ve-degree methodology:(7)χveYmn,n=∑x1x2∈Eξvex1+ξvex2−1/2=3n110+6n113+6n117+9n24−9n2+18mn+9118=32n28+31010+61313+61717+26−92+18mn+322.


## 4. Theve-Degree Results of Y-Junction Graph Ym1n,n

By attaching the2n pendants vertices with 2 degree vertices to any one tube of Ymn,n graph, we obtain a new graph, it is denoted by Ym1n,n, see Figure 2. The order and size of Ym1n,n graph is 3n2/2+11n+12mn+6 and 9n2/4+25n/2+18mn+9, respectively. This section determinen the ve-degree results of Y-junction graph Ym1n,n. The edge partition of end vertices ve-degree of each edge along with the degree of end vertices of each edge for Ym1n,n graph is given in Table 2.Figure 2
A variant ofY-junction Ym1n,n.Table 2
The end verticesve-degrees of each edge of Ym1n,n.
ξx1,ξx2ξvex1,ξvex2Count2,25,52n2,35,84n3,37,92n1,33,72n2,38,94n3,37,7n3,39,99n2/4−5n/2+18mn+9


### 4.1. The Firstve-Degree Zagreb β Index

(8)Mβve1Ym1n,n=∑x1x2∈Eξvex1+ξvex2=102n+134n+162n+102n+174n+14n+18+9n24−5n2+18mn+9=81n22+161+324mn+162.


### 4.2. The Secondve-Degree Zagreb Index

(9)Mve2Ym1n,n=∑x1x2∈Eξvex1×ξvex2,Mve2Ym1n,n=252n+404n+632n+212n+724n+49n+81+9n24−5n2+18mn+9=729+729n24+10252+1458mn.


### 4.3. The Randić Index Developed byve-Degree Methodology

Utilizing the edge-partition details described in the Table2, we measured the Randić index developed by ve-degree methodology:(10)RveYm1n,n=∑x1x2∈Eξvex1×ξvex2−1/2=35−1/22n+40−1/24n+63−1/22n+21−1/22n+72−1/24n+49−1/2n+81−1/29n24−5n2+18mn+9=1+n24+167630+105+2721+22121+23+2mn.


### 4.4. The Atom-Bond Connectivity Index Developed byve-Degree Methodology

Utilizing the edge-partition details described in the Table2, we measured the atom-bond connectivity index developed by ve-degree methodology:(11)ABCveYm1n,n=∑x1x2∈Eξvex1+ξvex2−2ξvex1×ξvex21/2=2n825+4n1140+2n1463+2n821+4n1572+n1249+16819n24−5n2+18mn+9=4+n2+22215+1105+44221+303+237−109+8mn.


### 4.5. The Geometric-Arithmetic Index Developed byve-Degree Methodology

Utilizing the edge-partition details described in the Table2, we measured the geometric-arithmetic index developed by ve-degree methodology:(12)GAveYm1n,n=∑x1x2∈E2ξvex1×ξvex21/2ξvex1+ξvex2=2n22510+4n24013+2n26316+2n22110+4n27217+n24914+281189n24−5n2+18mn+9=9+9n24+12+161013+374+2215+48217+18mn.


### 4.6. The Harmonic Index Developed byve-Degree Methodology

Utilizing the edge-partition details described in the Table2, we measured the harmonic index developed by ve-degree methodology:(13)HveYm1n,n=∑x1x2∈E2ξvex1+ξvex2=2n210+4n213+2n216+2n210+4n217+n214+2189n24−5n2+18mn+9=1+n24+557213278460+2mn.


### 4.7. The Sum-Connectivity Index Developed byve-Degree Methodology

Utilizing the edge-partition details described in the Table2, we measured the sum-connectivity index developed by ve-degree methodology:(14)χveYm1n,n=∑x1x2∈Eξvex1+ξvex2−1/2=2n110+4n113+2n116+2n110+4n117+n114+1189n24−5n2+18mn+9=2105+41313+12+41717+1414+26−52+18mn+32n28+322.


## 4.1. The Firstve-Degree Zagreb β Index

(8)Mβve1Ym1n,n=∑x1x2∈Eξvex1+ξvex2=102n+134n+162n+102n+174n+14n+18+9n24−5n2+18mn+9=81n22+161+324mn+162.


## 4.2. The Secondve-Degree Zagreb Index

(9)Mve2Ym1n,n=∑x1x2∈Eξvex1×ξvex2,Mve2Ym1n,n=252n+404n+632n+212n+724n+49n+81+9n24−5n2+18mn+9=729+729n24+10252+1458mn.


## 4.3. The Randić Index Developed byve-Degree Methodology

Utilizing the edge-partition details described in the Table2, we measured the Randić index developed by ve-degree methodology:(10)RveYm1n,n=∑x1x2∈Eξvex1×ξvex2−1/2=35−1/22n+40−1/24n+63−1/22n+21−1/22n+72−1/24n+49−1/2n+81−1/29n24−5n2+18mn+9=1+n24+167630+105+2721+22121+23+2mn.


## 4.4. The Atom-Bond Connectivity Index Developed byve-Degree Methodology

Utilizing the edge-partition details described in the Table2, we measured the atom-bond connectivity index developed by ve-degree methodology:(11)ABCveYm1n,n=∑x1x2∈Eξvex1+ξvex2−2ξvex1×ξvex21/2=2n825+4n1140+2n1463+2n821+4n1572+n1249+16819n24−5n2+18mn+9=4+n2+22215+1105+44221+303+237−109+8mn.


## 4.5. The Geometric-Arithmetic Index Developed byve-Degree Methodology

Utilizing the edge-partition details described in the Table2, we measured the geometric-arithmetic index developed by ve-degree methodology:(12)GAveYm1n,n=∑x1x2∈E2ξvex1×ξvex21/2ξvex1+ξvex2=2n22510+4n24013+2n26316+2n22110+4n27217+n24914+281189n24−5n2+18mn+9=9+9n24+12+161013+374+2215+48217+18mn.


## 4.6. The Harmonic Index Developed byve-Degree Methodology

Utilizing the edge-partition details described in the Table2, we measured the harmonic index developed by ve-degree methodology:(13)HveYm1n,n=∑x1x2∈E2ξvex1+ξvex2=2n210+4n213+2n216+2n210+4n217+n214+2189n24−5n2+18mn+9=1+n24+557213278460+2mn.


## 4.7. The Sum-Connectivity Index Developed byve-Degree Methodology

Utilizing the edge-partition details described in the Table2, we measured the sum-connectivity index developed by ve-degree methodology:(14)χveYm1n,n=∑x1x2∈Eξvex1+ξvex2−1/2=2n110+4n113+2n116+2n110+4n117+n114+1189n24−5n2+18mn+9=2105+41313+12+41717+1414+26−52+18mn+32n28+322.


## 5. Theve-Degree Results of Y-Junction Graph Ym2n,n

By attaching the4n pendants vertices with 2 degree vertices to any two tube of Ymn,n graph, we obtain a new graph, it is denoted by Ym2n,n, see Figure 3. The cardinality of Ym2n,n is 3n2/2+13n+12mn+6 and size is 9n2/4+29n/2+18mn+9. This section determined the ve-degree results of Y-junction graph Ym2n,n. The edge partition of end vertices ve-degree of each edge along with the degree of end vertices of each edge for Ym2n,n graph is given in Table 3.Figure 3
A variant ofY-junction Ym2n,n.Table 3
Theve-degrees of each edge of Ym2n,n.
ξx1,ξx2ξvex1,ξvex2Count2,25,5n2,35,82n3,37,94n1,33,74n2,38,92n3,37,72n3,39,99n2/4−n/2+18mn+9


### 5.1. The Firstve-Degree Zagreb β Index

(15)Mβve1Ym2n,n=∑x1x2∈Eξvex1+ξvex2=10n+132n+164n+104n+172n+142n+189n24−n2+18mn+9=162+81n22+193+324mn.


### 5.2. The Second Zagreb Index Developed byve-Degree Methodology

(16)Mve2Ym2n,n=∑x1x2∈Eξvex1×ξvex2=25n+402n+634n+214n+722n+492n+819n24−n2+18mn+9=729+729n24+12852+1458mn.


### 5.3. The Randić Index Developed byve-Degree Methodology

Utilizing the edge-partition details described in the Table3, we measured the Randić index developed by ve-degree methodology:(17)RveYm2n,n=∑x1x2∈Eξvex1×ξvex2−1/2=35−1/2n+40−1/22n+63−1/24n+21−1/24n+72−1/22n+49−1/22n+81−1/29n24−n2+18mn+9=1+n24+271630+1010+4721+42121+26+2mn.


### 5.4. The Atom-Bond Connectivity Index Developed byve-Degree Methodology

Utilizing the edge-partition details described in the Table3, we measured the atom-bond connectivity index developed by ve-degree methodology:(18)ABCveYm2n,n=∑x1x2∈Eξvex1+ξvex2−2ξvex1×ξvex21/2=n825+2n1140+4n1463+4n821+2n1572+2n1249+16819n24−n2+18mn+9=4+n2+26215+11010+84221+306+437−29+8mn.


### 5.5. The Geometric-Arithmetic Index Developed byve-Degree Methodology

Utilizing the edge-partition details described in the Table3, we measured the geometric-arithmetic index developed by ve-degree methodology:(19)GAveYm2n,n=∑x1x2∈E2ξvex1×ξvex21/2ξvex1+ξvex2=n22510+2n24013+4n26316+4n22110+2n27217+2n24914+281189n24−n2+18mn+9=9+9n24+52+81013+372+4215+24217+18mn.


### 5.6. The Harmonic Index Developed byve-Degree Methodology

Utilizing the edge-partition details described in the Table3, we measured the harmonic index developed by ve-degree methodology:(20)HveYm2n,n=∑x1x2∈E2ξvex1+ξvex2=n210+2n213+4n216+4n210+2n217+2n214+2189n24−n2+18mn+9=1+n24+3164913923+2mn.


### 5.7. The Sum-Connectivity Index Developed byve-Degree Methodology

Utilizing the edge-partition details described in the Table3, we measured the sum-connectivity index developed by ve-degree methodology:(21)χveYm2n,n=∑x1x2∈Eξvex1+ξvex2−1/2=n110+2n113+4n116+4n110+2n117+2n114+1189n24−n2+18mn+9=102+21313+1+21717+147+26−12+18mn+32n28+322.


## 5.1. The Firstve-Degree Zagreb β Index

(15)Mβve1Ym2n,n=∑x1x2∈Eξvex1+ξvex2=10n+132n+164n+104n+172n+142n+189n24−n2+18mn+9=162+81n22+193+324mn.


## 5.2. The Second Zagreb Index Developed byve-Degree Methodology

(16)Mve2Ym2n,n=∑x1x2∈Eξvex1×ξvex2=25n+402n+634n+214n+722n+492n+819n24−n2+18mn+9=729+729n24+12852+1458mn.


## 5.3. The Randić Index Developed byve-Degree Methodology

Utilizing the edge-partition details described in the Table3, we measured the Randić index developed by ve-degree methodology:(17)RveYm2n,n=∑x1x2∈Eξvex1×ξvex2−1/2=35−1/2n+40−1/22n+63−1/24n+21−1/24n+72−1/22n+49−1/22n+81−1/29n24−n2+18mn+9=1+n24+271630+1010+4721+42121+26+2mn.


## 5.4. The Atom-Bond Connectivity Index Developed byve-Degree Methodology

Utilizing the edge-partition details described in the Table3, we measured the atom-bond connectivity index developed by ve-degree methodology:(18)ABCveYm2n,n=∑x1x2∈Eξvex1+ξvex2−2ξvex1×ξvex21/2=n825+2n1140+4n1463+4n821+2n1572+2n1249+16819n24−n2+18mn+9=4+n2+26215+11010+84221+306+437−29+8mn.


## 5.5. The Geometric-Arithmetic Index Developed byve-Degree Methodology

Utilizing the edge-partition details described in the Table3, we measured the geometric-arithmetic index developed by ve-degree methodology:(19)GAveYm2n,n=∑x1x2∈E2ξvex1×ξvex21/2ξvex1+ξvex2=n22510+2n24013+4n26316+4n22110+2n27217+2n24914+281189n24−n2+18mn+9=9+9n24+52+81013+372+4215+24217+18mn.


## 5.6. The Harmonic Index Developed byve-Degree Methodology

Utilizing the edge-partition details described in the Table3, we measured the harmonic index developed by ve-degree methodology:(20)HveYm2n,n=∑x1x2∈E2ξvex1+ξvex2=n210+2n213+4n216+4n210+2n217+2n214+2189n24−n2+18mn+9=1+n24+3164913923+2mn.


## 5.7. The Sum-Connectivity Index Developed byve-Degree Methodology

Utilizing the edge-partition details described in the Table3, we measured the sum-connectivity index developed by ve-degree methodology:(21)χveYm2n,n=∑x1x2∈Eξvex1+ξvex2−1/2=n110+2n113+4n116+4n110+2n117+2n114+1189n24−n2+18mn+9=102+21313+1+21717+147+26−12+18mn+32n28+322.


## 6. Theve-Degree Results of Y-Junction Graph Ym3n,n

InYmn,n when one tube appears with exactly 2n pendants, we denote it by Ym1n,n, see Figure 2. The order and size of this new graph is 3n2/2+11n+12mn+6 and 9n2/4+25n/2+18mn+9, respectively. The Y-junction graph Ym2n,n is obtained by attaching 4n pendants to any two tubes of Ymn,n, see Figure 3. The cardinality of Ym2n,n is 3n2/2+13n+12mn+6 and size is 9n2/4+29n/2+18mn+9. The graph Ymn,n with maximum possible pendants denoted by Ym3n,n, see Figure 4. It has order 3n2/2+15n+12mn+6 and size 9n2/4+33n/2+18mn+9.Figure 4
A variant of Y-junctionYm3n,n.


### 6.1. The Firstve-Degree Zagreb β Index

(22)Mβve1Ym3n,n=∑x1x2∈Eξvex1+ξvex2=166n+106n+143n+189n24+3n2+18mn+9=162+81n22+225+324mn.


### 6.2. The Second Zagreb Index Developed byve-Degree Methodology

(23)Mve2Ym3n,n=∑x1x2∈Eξvex1×ξvex2=636n+216n+493n+819n24+3n2+18mn+9=729+729n24+15452+1458mn.


### 6.3. The Randić Index Developed byve-Degree Methodology

Utilizing the edge-partition details described in the Table4, we measured the Randić index developed by ve-degree methodology:(24)RveYm3n,n=∑x1x2∈Eξvex1×ξvex2−1/2=63−1/26n+21−1/26n+49−1/23n+81−1/29n24+3n2+18mn+9=1+n24+277+2217+2542+2mn.Table 4
Theve-degrees of each edge of Ym3n,n.
ξx1,ξx2ξvex1,ξvex2Count3,37,96n1,33,76n3,37,73n3,39,99n2/4−3n/2+18mn+9


### 6.4. The Atom-Bond Connectivity Index Developed byve-Degree Methodology

Utilizing the edge-partition details described in the Table4, we measured the atom-bond connectivity index developed by ve-degree methodology:(25)ABCveYm3n,n=∑x1x2∈Eξvex1+ξvex2−2ξvex1×ξvex21/2=6n1463+6n821+3n1249+16819n24+3n2+18mn+9=4+n2+22+4427+637+23+8mn.


### 6.5. The Geometric-Arithmetic Index Developed byve-Degree Methodology

Utilizing the edge-partition details described in the Table4, we measured the geometric-arithmetic index developed by ve-degree methodology:(26)GAveYm3n,n=∑x1x2∈E2ξvex1×ξvex21/2ξvex1+ξvex2=6n26316+6n22110+3n24914+281189n24+3n2+18mn+9=9+9n24+974+6215+92+18mn.


### 6.6. The Harmonic Index Developed byve-Degree Methodology

Utilizing the edge-partition details described in the Table4, we measured the harmonic index developed by ve-degree methodology:(27)HveYm3n,n=∑x1x2∈E2ξvex1+ξvex2=6n216+6n210+3n214+2189n24+3n2+18mn+9=1+n24+1069420+2mn.


### 6.7. The Sum-Connectivity Index Developed byve-Degree Methodology

Utilizing the edge-partition details described in the Table4, we measured the sum-connectivity index developed by ve-degree methodology:(28)χveYm3n,n=∑x1x2∈Eξvex1+ξvex2−1/2=6n116+6n110+3n114+1189n24+3n2+18mn+9=32n28+32+3105+31414+2632+18mn+322.


## 6.1. The Firstve-Degree Zagreb β Index

(22)Mβve1Ym3n,n=∑x1x2∈Eξvex1+ξvex2=166n+106n+143n+189n24+3n2+18mn+9=162+81n22+225+324mn.


## 6.2. The Second Zagreb Index Developed byve-Degree Methodology

(23)Mve2Ym3n,n=∑x1x2∈Eξvex1×ξvex2=636n+216n+493n+819n24+3n2+18mn+9=729+729n24+15452+1458mn.


## 6.3. The Randić Index Developed byve-Degree Methodology

Utilizing the edge-partition details described in the Table4, we measured the Randić index developed by ve-degree methodology:(24)RveYm3n,n=∑x1x2∈Eξvex1×ξvex2−1/2=63−1/26n+21−1/26n+49−1/23n+81−1/29n24+3n2+18mn+9=1+n24+277+2217+2542+2mn.Table 4
Theve-degrees of each edge of Ym3n,n.
ξx1,ξx2ξvex1,ξvex2Count3,37,96n1,33,76n3,37,73n3,39,99n2/4−3n/2+18mn+9


## 6.4. The Atom-Bond Connectivity Index Developed byve-Degree Methodology

Utilizing the edge-partition details described in the Table4, we measured the atom-bond connectivity index developed by ve-degree methodology:(25)ABCveYm3n,n=∑x1x2∈Eξvex1+ξvex2−2ξvex1×ξvex21/2=6n1463+6n821+3n1249+16819n24+3n2+18mn+9=4+n2+22+4427+637+23+8mn.


## 6.5. The Geometric-Arithmetic Index Developed byve-Degree Methodology

Utilizing the edge-partition details described in the Table4, we measured the geometric-arithmetic index developed by ve-degree methodology:(26)GAveYm3n,n=∑x1x2∈E2ξvex1×ξvex21/2ξvex1+ξvex2=6n26316+6n22110+3n24914+281189n24+3n2+18mn+9=9+9n24+974+6215+92+18mn.


## 6.6. The Harmonic Index Developed byve-Degree Methodology

Utilizing the edge-partition details described in the Table4, we measured the harmonic index developed by ve-degree methodology:(27)HveYm3n,n=∑x1x2∈E2ξvex1+ξvex2=6n216+6n210+3n214+2189n24+3n2+18mn+9=1+n24+1069420+2mn.


## 6.7. The Sum-Connectivity Index Developed byve-Degree Methodology

Utilizing the edge-partition details described in the Table4, we measured the sum-connectivity index developed by ve-degree methodology:(28)χveYm3n,n=∑x1x2∈Eξvex1+ξvex2−1/2=6n116+6n110+3n114+1189n24+3n2+18mn+9=32n28+32+3105+31414+2632+18mn+322.


## 7. Conclusion

In this research work,ve-degree topological indices are measured of Y-junctions and their three different variants. We determined the first ve-degree Zagreb β-index, second Zagreb index, Randić, atom-bond-connectivity index, general sum-connectivity and geometric-arithmetic, and harmonic index developed by ve-degree methodology, for four types of Y-shaped carbon nanotube junctions Ymn,n. The results of Y-junctions and their structures also are elaborated in numerical Tables 5–8. Instead of a whole complex structure, it will be easy to see as a numeric quantity.Table 5
Numerical comparison ofMβve1, Mve2, Hve, Rve, χve, ABCve, GAve for Y-junction graph Ymn,n.
m,nMβve1Mve2RveABCveGAveHveχve(5, 5)9919.5043647.866.0289256.910566.88865.8948136.481(6, 6)14058.062073.092.5347361.493799.96692.3738191.992(7, 7)18925.583778.8123.541484.0741073.55123.353257.048(8, 8)24522.0108765.0159.046624.6561387.62158.832331.649(9, 9)30847.5137032.0199.052783.2401742.20198.811415.797(10, 10)37902.0168579.0243.558959.8202137.28243.290509.491(11, 11)45685.5203407.0292.5641154.402572.86292.269612.730(12, 12)54198.0241515.0346.0691366.983048.94345.748725.515(13, 13)63439.5282904.0404.0751597.563565.50403.726847.847(14, 14)73410.0327573.0466.5811846.154122.58466.205979.724Table 6
Numerical comparison ofMβve1, Mve2, Hve, Rve, χve, ABCve, GAve for Y-junction graph Ym1n,n.
m,nMβve1Mve2RveABCveGAveHveχve(5, 5)10079.544297.867.5368262.078576.26267.2553139.058(6, 6)14250.062853.094.3441367.695811.21494.0063195.082(7, 7)19149.584688.8125.652491.3091086.66125.257260.653(8, 8)24778.0109805.0161.459632.9251402.61161.008335.770(9, 9)31135.5138202.0201.767792.5431759.08201.259420.433(10, 10)38222.0169879.0246.574970.1572156.02246.011514.641(11, 11)46037.5204837.0295.8811165.772593.47295.262618.397(12, 12)54582.0243075.0349.6881379.393071.43349.013731.697(13, 13)63855.5284594.0407.9961611.03589.89407.264854.543(14, 14)73858.0329393.0470.8031860.624148.83470.015986.937Table 7
Numerical comparison ofMβve1, Mve2, Hve, Rve, χve, ABCve, GAve for Y-junction graph Ym2n,n.
m,nMβve1Mve2RveABCveGAveHveχve(5, 5)10239.544947.869.0446267.247585.63668.6157141.634(6, 6)14442.063633.096.1536373.896822.46395.6389198.173(7, 7)19373.585598.8127.763498.5451099.79127.162264.260(8, 8)25034.0110845.0163.872641.1941417.62163.185339.891(9, 9)31423.5139372.0204.480801.8451775.94203.708425.069(10, 10)38542.0171179.0249.590980.4942174.78248.731519.793(11, 11)46389.5206267.0299.1991177.152614.10298.255624.062(12, 12)54966.0244635.0353.3061391.803093.92352.278737.878(13, 13)64271.5286284.0411.9151624.453614.25410.801861.241(14, 14)74306.0331213.0475.0241875.094175.07473.824994.147Table 8
Numerical comparison ofMβve1, Mve2, Hve, Rve, χve, ABCve, GAve for Y-junction graph Ym3n,n.
m,nMβve1Mve2RveABCveGAveHveχve(5, 5)10399.545597.870.5523272.414595.01069.9762144.209(6, 6)14634.064413.097.9629380.098833.71397.2714201.264(7, 7)19597.586508.8129.873505.7811112.91129.067267.865(8, 8)25290.0111885.0166.283649.4631432.61165.362344.012(9, 9)31711.5140542.0207.194811.1481792.82206.157429.705(10, 10)38862.0172479.0252.604990.8302193.52251.452524.945(11, 11)46741.5207697.0302.5151188.512634.72301.248629.729(12, 12)55350.0246195.0356.9261404.203116.43355.543744.059(13, 13)64687.5287974.0415.8361637.883638.63414.338867.937(14, 14)74754.0333033.0479.2461889.564201.33477.6331001.36



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*Source: 2383074-2022-07-20.xml*