Study of Vanadium Carbide Structures Based on Ve and Ev-Degree Topological Indices

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Introduction
Vertex degree concept has devised many topological indices that are applicable in QSPR/QSAR studies.Topological indices are widely used in theoretical and mathematical chemistry as they are associated with the topology of a chemical structure along with its other identical properties such as boiling points, strain energy, and stability [1].In chemical graph theory, a chemical graph is referred as a molecular structure with atoms as its vertices and chemical bonds as its edges.A topological index is a numerical parameter that creates a link between the physical and chemical properties of a molecule [2].Many topological descriptors based on degree have been introduced.ese topological indices have provided assistance in calculating different parametric calculations related to molecular structures to make them understandable and beneficial.A lot of topological descriptors have been defined and studied so far, but Zagreb indices [3], Weiner index [4], and Randic index [5] are the most studied among all of them.To read more about the chemical applicability of topological descriptors, see [5][6][7][8][9][10][11][12].
Researchers have attempted to study the varying behavior of transition metal carbides due to their complex structures.Such mineral metals are available in commercial places, and their salts are broadly utilized in our enterprises related to electrochemistry and material science.Among these, vanadium carbide complexes have shown crystal morphologies and stoichiometrics and display a great variety of superstructures.Very recently, many attempts have been done to purify high quality vanadium carbide by presenting different binary model system such as VC, V 2 C, V 4 C 3 , V 6 C 5 , and V 8 C 7 .For more details, see [13][14][15][16].
Let G be a simple connected graph with its edge set and vertex set denoted by E and V, respectively.e neighbor set N(v) of a vertex v contains those vertices v 1 such that vv 1 ∈ E. e degree of vertex v is denoted by d v and is the cardinality of the set N(v).
∪ v { } be the closed neighborhood of v. To read more about the basic concepts related to graph, see [17].
M. Chellali et al. [18] first introduced the concept of ev degree of an edge e and ve degree of a vertex v. e ev degree of an edge e is denoted by d ev (e) and is defined as the total number of vertices in the closed neighborhoods of the end vertices of an edge e. e ve degree of a vertex v is denoted by d ve (v) and is the total number of edges that are adjacent with v and the first neighbor of v. Ediz [3] first introduced the concept of ve degree and ev degree Zagreb and Randic indices.
e mathematical formulas of these indices are presented in Table 1.
ese newly defined indices were compared with Zagreb, Weiner, and Randic indices by modeling some of the physical/chemical properties of octane isomers.ese indices have been observed to provide better correlation than the Randic, Weiner, and Zagreb indices for predicting some specific physical and chemical properties of octane isomers.Recently, a lot of work is done in the direction of computing newly defined ve degree and ev degreebased indices [19][20][21][22][23].

Vanadium Carbide
Vanadium carbide belongs to the family of group IV to VI transition metal carbides and shows homogeneity to metal nitrides, monocarbides, and carbonitrides.ey possess unique associations between physio-chemical properties such as high melting points, high temperature resistivity, strength, and hardness which are associated with good electrical and thermal conductivity.ese rare combinations of properties make such compounds very interesting for the researchers.ese materials can be used as wear-resistant hard alloys and as hard coatings for protection purposes, due to their nanochemical properties [24,25].
Vanadium carbide is the hardest inorganic metal-carbide with the formula VC.VC is an incredibly hard refractory ceramic with exceptional wear resistance, high modulus of elasticity (400 GPa), and good strength retention even at high temperatures [26][27][28].VC coatings are used in corrosion prevention, cutting tool application, machining, drilling, and dyeing.Some industrial uses of VC are given in [27,[29][30][31].We denote the crystallographic structure of vanadium carbide by VC [m, n]. e molecular structure of vanadium carbide for m � 5 � n is depicted in Figure 1. e structure of VC[m, n] has total number of 3mn + m + n vertices and 6mn − m − n edges.Let V i denote the vertex set containing the vertices of VC[m, n] of degree i. en, the vertex set V(VC [m, n]) can be partitioned into six sets with

Main Results
Theorem 1.Let m, n ≥ 2, then Proof.To compute the ev degree Zagreb and ev degree Randic index of VC[m, n], we need to compute the ev degree of the edges in each partition set E (i,j) . is calculation is presented in Table 2. Now, using the information presented in Table 2 and the definition of ev degree Zagreb and ev degree Randic index, we get � (5)    Table 2: Ev-degrees of edges of vanadium carbide.
Journal of Chemistry

Numerical Results and Discussion
Topological indices are used as vital tools for the analysis of chemicals, given the essential topology of chemical structures.Zagreb-type indices are used to calculate the total π− electronic energy of molecules [32].e Randich index is commonly used to determine the chemical similarity of molecular compounds, as well as to calculate the boiling point and Kovaz constants of molecules [33].e atom-bond connectivity index (ABC) provides a very good correlation for calculating strain energies as well as for the stability of linear and branched chemical structures [34].It can be seen from Table 4 and         Journal of Chemistry

Conclusion
Graphs invariants are calculated by some well-known topological indices which are important tools for resembling and forecasting the properties of chemical compounds in QSPRs and the QSARs.e TI is a numerical measure that represents the biological, physical, and chemical properties of molecules such as boiling, melting, and flickering point; moisture; and forming heat.In this paper, we have computed the ev degree and ve degree-based topological indices with graphical representations for the molecular structure of vanadium carbide for a better understanding of pharmaceutical, physical, chemical, and biological properties.
5 and Figures 2-6 of indices, an increase in the value of m and n raises the values of topological descriptors for vanadium carbide structure.

Table 1 :
Mathematical formula of topological indices.

Table 3 :
Edge Partition of vanadium carbide.

Table 4 :
Numerical results of indices for vanadium carbide.

Table 5 :
Numerical results of indices for vanadium carbide.