Abstract
Vanadium is a biologically active product with significant industrial and biological applications. Vanadium is found in a variety of minerals and fossil fuels, the most common of which are sandstones, crude oil, and coal. Topological descriptors are numerical numbers assigned to the molecular structures and have the ability to predict certain of their physical/chemical properties. In this paper, we have studied topological descriptors of vanadium carbide structure based on ev and ve degrees. In particular, we have computed the closed forms of Zagreb, Randic, geometric-arithmetic, and atom-bond connectivity (ABC) indices of vanadium carbide structure based on ev and ve degrees. This kind of study may be useful for understanding the biological and chemical behavior of the structure.
1. 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. These 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 [5–12].
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 , and . For more details, see [13–16].
Let be a simple connected graph with its edge set and vertex set denoted by and , respectively. The neighbor set of a vertex contains those vertices such that . The degree of vertex is denoted by and is the cardinality of the set . Let be the closed neighborhood of . 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 and ve degree of a vertex . The ev degree of an edge is denoted by and is defined as the total number of vertices in the closed neighborhoods of the end vertices of an edge . The ve degree of a vertex is denoted by and is the total number of edges that are adjacent with and the first neighbor of . Ediz [3] first introduced the concept of ve degree and ev degree Zagreb and Randic indices. The mathematical formulas of these indices are presented in Table 1. These newly defined indices were compared with Zagreb, Weiner, and Randic indices by modeling some of the physical/chemical properties of octane isomers. These 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 degree-based indices [19–23].
2. 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. They 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. These rare combinations of properties make such compounds very interesting for the researchers. These 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 is an incredibly hard refractory ceramic with exceptional wear resistance, high modulus of elasticity (400 GPa), and good strength retention even at high temperatures [26–28]. VC coatings are used in corrosion prevention, cutting tool application, machining, drilling, and dyeing. Some industrial uses of VC are given in [27, 29–31]. We denote the crystallographic structure of vanadium carbide by . The molecular structure of vanadium carbide for is depicted in Figure 1. The structure of has total number of vertices and edges. Let denote the vertex set containing the vertices of of degree . Then, the vertex set can be partitioned into six sets with , , , , , and . Let denote the edge set containing the edges of with end vertices of degree and degree . The edge set of can be partitioned based on the degree of end vertices as follows: with 1 edge, with 2 edges, with edges, with 2 edges, with edges, with edges, with 1 edge, with edges, and having edges. In Theorem 1, we compute the ev degree Randic and ev degree Zagreb index of .
3. Main Results
Theorem 1. Let , then
Proof. To compute the ev degree Zagreb and ev degree Randic index of , we need to compute the ev degree of the edges in each partition set . This 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
Theorem 2. Let , then
Proof. To compute the ve degree-based indices, we need to find the edge partition of based on the ve degree of end vertices of each edge. This partition is presented in Table 3. Now, using the values from Table 3 and the definition of ve degree indices, we get
4. 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 electronic energy of molecules [32]. The 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]. The 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 5 and Figures 2–6 of indices, an increase in the value of and raises the values of topological descriptors for vanadium carbide structure.
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5. 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. The 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.
Data Availability
No data were used to support this study.
Disclosure
This research was carried out as a part of employment of the authors.
Conflicts of Interest
The authors hereby declare that there are no conflicts of interest regarding the publication of this paper.