Comparative Study of Y-Junction Nanotubes with Vertex-Edge Based Topological Descriptors

e current results of various forms of carbon nanostructures and its applications in dierent areas attract the researchers. In pharmaceutical, medicine, industry and electronic devices they used it by its graphical invariants. e detection of dierent types of carbon nanotubes junctions enhanced the attention and interest for forthcoming devices like transistors and ampliers. 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 of ve-degree topological indices for various type of carbon nanotubes Y-junctions and their comparisons. e particular indices called as e rst 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.


Introduction
Let a graph having vertex set V and edge set E possesses the properties of connectivity, usually labeled as G (V, E). For a vertex x 1 ∈ V, the concept of open neighborhood of that vertex x 1 is formulated as N(x 1 ) x 2 ∈ V: x 1 x 2 ∈ E , while the concept of closed neighborhood formulated and notated by N[x 1 ] N(x 1 ) ∪ x 1 , [1][2][3]. A notation ξ ve (x 1 ), is used for the ve-degree of any vertex x 1 ∈ V, and measured by the count of distinct edges which are incident to any vertex from the closed neighborhood of x 1 . Further detail and discussion on this notation and its mathematical de nition, one can see [4][5][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, threeterminal devices and carbon nanotube graphs have tremendous potential. While study the chemical things for various determinations in di erent areas, the energy bond is the one of the most important thermophysical to be measured. ere are di erent type of nanotubes junctions for example, X, Y, L and T and their applications can be seen in [9][10][11][12].
e 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][14][15][16]. e 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 di erent types of topological indices, see [20][21][22][23][24][25]. Some new variants and generalized results on the topological descriptors are found in the articles suggested [3,26,27].
ere are variety of topological descriptors, one of them is the vertex-edge based that will be discussed in this article. e researchers in [1], de ned the "ev-degree," and [4] contributed in this study. Basic de nitions regarding "ve-degree" topological indices, refer to [28]. e vertex-edge based topological descriptors are: e rst ve-degree Zagreb β index (M 1 βve (Y m (n, n)) 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][32][33][34][35][36][37]. In this paper, the exact values of vertexedge based topological indices for Y-junctions carbon nanotubes are determined.
In this work, a junction graph labeled with Y m (n, n) is graphs having no pendent or degree one vertex, exists. is work also consists of other three topologies of Y-junction graphs labeled with Y 1 m (n, n), Y 2 m (n, n) and Y 3 m (n, n) and these contained some vertices with degree one. ese further topologies are constructed by Y m (n, n)-junction graphs by adding pendants to degree 2 vertices. Single tube among three tubes of Y m (n, 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 Y m (n, n) and 2n for each tube.

The ve-Degree Results of Y-Junction
Graph Y m (n, n) is section presented the ve-degree results of Y-junction graph Y m (n, n). is graph does not contain any pendent vertex that is shown in Figure 1. e edge partition of end vertices ve-degree of each edge along with the degree of end vertices of each edge for Y m (n, n) graph is given in Table 1.

e Randić Index Developed by ve-Degree Methodology.
Utilizing the edge-partition details described in the Table 1, we measured the Randić index developed by ve-degree methodology:

e Atom-Bond Connectivity Index Developed by ve-Degree
Methodology. Utilizing the edge-partition details described in the Table 1, we measured the atom-bond connectivity index developed by ve-degree methodology:

e Geometric-Arithmetic Index Developed by ve-Degree
Methodology. Utilizing the edge-partition details described in the Table 1, we measured the geometric-arithmetic index developed by ve-degree methodology:

e Harmonic Index Developed by ve-Degree
Methodology. Utilizing the edge-partition details described in the Table 1, we measured the harmonic index developed by ve-degree methodology: Journal of Mathematics

3.7.
e Sum-Connectivity Index Developed by ve-Degree Methodology. Utilizing the edge-partition details described in the Table 1, we measured the sum-connectivity index developed by ve-degree methodology:
is section determinen the ve-degree results of Y-junction graph Y 1 m (n, n). e edge partition of end vertices ve-degree of each edge along with the degree of end vertices of each edge for Y 1 m (n, n) graph is given in Table 2.

e Randić Index Developed by ve-Degree Methodology.
Utilizing the edge-partition details described in the Table 2, we measured the Randić index developed by ve-degree methodology:

e Geometric-Arithmetic Index Developed by ve-Degree
Methodology. Utilizing the edge-partition details described in the Table 2, we measured the geometric-arithmetic index developed by ve-degree methodology:

e Sum-Connectivity Index Developed by ve-Degree
Methodology. Utilizing the edge-partition details described in the Table 2, we measured the sum-connectivity index developed by ve-degree methodology:

The ve-Degree Results of Y-Junction
Graph Y 2 m (n, n) By attaching the 4n pendants vertices with 2 degree vertices to any two tube of Y m (n, n) graph, we obtain a new graph, it is denoted by Y 2 m (n, n), see Figure 3. e cardinality of Y 2 m (n, n) is (3n 2 /2) + 13n + 12mn + 6 and size is (9n 2 /4) + (29n/2) + 18mn + 9. is section determined the ve-degree results of Y-junction graph Y 2 m (n, n). e edge partition of end vertices ve-degree of each edge along with the degree of end vertices of each edge for Y 2 m (n, n) graph is given in Table 3.

e Randić Index Developed by ve-Degree Methodology.
Utilizing the edge-partition details described in the Table 3, we measured the Randić index developed by ve-degree methodology:

e Geometric-Arithmetic Index Developed by ve-Degree
Methodology. Utilizing the edge-partition details described in the Table 3, we measured the geometric-arithmetic index developed by ve-degree methodology:     Table 3, we measured the harmonic index developed by ve-degree methodology:

e Sum-Connectivity Index Developed by ve-Degree
Methodology. Utilizing the edge-partition details described in the Table 3, we measured the sum-connectivity index developed by ve-degree methodology:

e Randić Index Developed by ve-Degree Methodology.
Utilizing the edge-partition details described in the Table 4, we measured the Randić index developed by ve-degree methodology:

e Atom-Bond Connectivity Index Developed by ve-Degree
Methodology. Utilizing the edge-partition details described in the Table 4, we measured the atom-bond connectivity index developed by ve-degree methodology:

e Geometric-Arithmetic Index Developed by ve-Degree
Methodology. Utilizing the edge-partition details described in the Table 4, we measured the geometric-arithmetic index developed by ve-degree methodology: 6.6. e Harmonic Index Developed by ve-Degree Methodology. Utilizing the edge-partition details described in the Table 4, we measured the harmonic index developed by ve-degree methodology: Table 4: e ve-degrees of each edge of Y 3 m (n, n).

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 Y m (n, n). e 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.

Data Availability
ere is not data associative with this manuscript.

Conflicts of Interest
e author declares that he has no conflicts of interest.