Research Article Study of Hardness of Superhard Crystals by Topological Indices

Topological indices give immense information about a molecular structure or chemical structure. The hardness of materials for the indentation can be deﬁned microscopically as the total resistance and eﬀect of chemical bonds in the respective materials. The aim of this paper is to study the hardness of some superhard BC x crystals by means of topological indices, speciﬁcally Randi´c index and atom-bond connectivity index.


Introduction
Hardness measures the crystal property of resistance into its deformation. In crystal-type materials, the resistence depends on the chemical bonds between its atoms. In the case of common metals and materials, there exists a delocalized form of bonding. For the domination of the hardness value, the dislocation density which is stored in metals is sufficient.
ere are many strategies that exist to establish the microscopic theory for the hardness level, and the main ideas are to analyze the experimental material (or metal). ere also exist some microscopic hardness models for the prediction of hardness, and these can be applied to covalent (and in some cases to ionic type) crystals (see Gao et al. [1]). A technique to relate the hardness of Vickers for a large class of crystals of covalent type to their microscopic properties has been studied in [2]. To energetically break an electron pair bond has the meaning that two electrons excite from the valence band to the conduction band in covalent crystals. In [3], Gilman proved that the activation energy that is required for a plastic slip is double the band gap denoted as E g . e force of resistance of a bond can be computed by studying respective E g of the materials. e form of the hardness for pure covalent-type crystals consists of three variables, and it is formulated as where A is the constant of proportionality and N a represents the covalent bond number, and it is per unit area; this area is such that it can be computed from the valence electron density N e as where n i is the position in the form of a number of the ith atom in the unit cell, Z i is the valence electron number of the ith atom crediting to the covalent bond, and V is the volume of the unit cell. More results on the hardness (various definitions of hardness) study of crystals can be referred to Suzuki et al. [4], Armstrong et al. [5], Mamun et al. [6], Shkir et al. [7], and Palatnikov et al. [8]. Now, if we talk about the unit crystals studied in this article, then the first one is BC 2 , and by the first-principles computations, BC 2 was predicted originally from (a kind of tetragonal phase) the cubic diamond structure. e lattice of the BC 2 structure is tetragonal, and this structure possesses the simple kind of the stacking sequence such as BC 2 BC 2 . . . along with the c-axis. Monitoring different states of the electronic densities of BC 2 shows that, in the crystal, all the B − Cand C − C-type bonds are purely metallic. e known hardness level of BC 2 is 56 GPa (gigapascal), and this level is very close to that of cubic boron nitride. Now, compound BC 3 has a different type of crystal structure which is more like and similar to graphite, and so, it has a hexagonal crystal-type structure. A study was performed in [2] to investigate some improved oxidation resistance in a graphitic material that contains very high collections of secondary boron. e procedure to create and find such samples is a reaction between boron trichloride and benzene to examine for chemical composition and crystal structure.
Another class of recently discovered compounds, which has almost the same structure as that of the diamond, is the crystal structure of BC 5 . For the purpose of correlation, compound BC 5 is very important, and it is also very useful to know under which type of conditions this compound can be extracted. As a function of pressure, the stability of compound BC 5 is relative to a solution of compound BC 3 and graphite.
Another similar but different compound is the BC 7 crystal structure which is more like to both compound structures of graphite and diamond. By computing the constants of elasticity and frequencies of phonon, the structural stability of the assumed compound crystal structure BC 7 has been confirmed. Similar to the direction for tensile strength of the diamond-like BC 7 , its ideal tensile strength was 155.2 GPa; this strength is about 52% more than that of the recent diamond-like predicted structure BC 5 . e theoretical Vickers hardness of the diamond crystals like BC 7 was 78 GP a which indicates that it is a superhard material; these readings show that BC 7 is a superhard material (see in [2]).
We can formulate the Vickers hardness in the form of f i , N e , and d as follows: where f i is the ionicity of the chemical bond in a crystal scale and d is the bond length in angstroms [9]. In the 70's, one of the famous degree-based indices is the Randić index which was introduced by Randić [10] in 1975, and it is characterized as In 1998, Bollobás and Erdos [11] and Amić et al. [12] proposed the general Randić index which is stated as Das et al. [13] studied the relationship between the Randić index and other degree-based indices. Milivojević and Pavlović [14] presented the extremal value and graphs for the variation of the Randić index with regard to minimum and maximum degrees.
Atom-bond connectivity index (in short, ABC index) was introduced by Estrada et al. [15] to measure the stability of alkanes and the strain energy of cycloalkanes which can be formulated by Dimitrov [16] provided an affirmative answer of a strengthened version of the previous conjecture and presented that a tree with a minimal ABC index cannot contain a pendent path of length 3 if its order is larger than 415. Dimitrov and Milosavljević [17] manifested several properties of the degree sequences of the trees with minimal ABC index, and a new algorithm for minimal-ABC trees is given. e definitions of more topological indices and results can be referred to Gao et al. [18][19][20][21][22]. Some more literature studies are also available in [23,24]. e applications of moving correlation coefficient technique enable us to examine the variation in the degree of correlation between correlation stratigraphic sequence and analysis the measure of variables within the framework of a single stratigraphic sequence. Cumulative correlation technique allows to determining more precisely, where such variation took place and it influences every member within the sequence of the preceding ones.
Since both hardness properties and topological indices are important topics in crystal science, it inspired us to study the relationship between them. e main contribution of this paper is to study the hardness of some superhard BC x crystals in the light of topological indices.

Main Results
Many B − C binary systems show high resistance to oxidation and reaction with ferrous metals, compared with the carbon-based materials. In Figure 1, we present selected BC x systems with specific crystal structures found in [2]. e Randić and atom-bond connectivity indices of these 12 types of BC crystals are computed as follows.
From Figure 1(a), we can see that the number of edges in the unit cell of BC 2 /41/amd is 36. We present the edge partition of BC 2 /41/amd in Table 1 based on the degree of vertices of each edge. e Randić index and ABC index for the BC 2 /41/amd crystal are computed using Table 1, and they are 14.136584 and 25.3452096, respectively. e number of edges of BC 3 P4 -m2, from Figure 1(b), is 12, and 8 of its edges are of type (1,3), and 4 are of type (3,4).
is gives us the Randić and ABC indices 5.773502692 and 9.113961545, respectively. e number of edges in both BC 3 Pmmaa and BC 3 Pmmab is 12 by Figures 1(c) and 1(d). e edge type of these two structures is also the same, containing 4 edges of type (1, 3), 2 edges of type (3, 3), 4 edges of type (2,3), and 2 edges of type (2,2). So, the Randić and ABC indices of         Number of obs = 12 Figure 9: Tabular form of the magnitude of the effectiveness of ABCI and HC over RI as shown in Figure 6.    Figure 1. e hardness of all the crystals is given in Table 2. Table 3 shows the crystal structures of Figure 1 along with their Randić index (R), atom-bond connectivity (ABC) index, calculated hardness (H) from Table 2, and cumulative correlations (Cor) between (R, H) and (ABC, H).

Comparison
In this section, we have compared the hardness of subjected materials with Randic and ABC indices. Figure 2 shows the comparison between the Randic index and hardness of the subjected materials given in Table 3. Figure 3 shows the comparison between the ABC index and hardness. Figure 4 shows the comparison between the Randic index and ABC index of the subjected materials given in Figure 1. Figure 5 shows the comparison between the hardness, Randic index, and ABC index of the subjected materials given in Figure 1.

Conclusion and Discussion
We have investigated the association among the Randić index (RI), hardness of the crystal (HC), and atom-bond connectivity index (ABCI). For this purpose, the structural equation model (SEM) has been applied by the structure of three equations. Figures 6-8 show us the magnitude of the effectiveness of ABCI and HC over RI, of RI and HC over ABCI, and of ABCI and RI over HC, respectively. Figures 9-11 give us the tabular form of the magnitude of the effectiveness of ABCI and HC over RI as shown in Figure 6, of RI and HC over ABCI as shown in Figure 7, and of ABCI and RI over HC as shown in Figure 8, respectively. Estimates of three equations depict that ABCI is positively and significantly associated with RI, and 1 unit increase in ABCI improves 0.90 units of RI, while HC is positively and insignificantly related with RI. Negative and insignificant magnitude of HC is observed with ABCI. However, RI is positively and significantly related with ABCI. Contradictory results are observed about the impact of RI and ABCI on HC; both RI and ABCI are insignificantly linked with HC. In this article, we have started a new study which relates the hardness and topological indices of superhard crystals; which can contribute to some futuristic applications, and we encourage others to do more research in this area.

Data Availability
All the data used to support the findings of this study are included within the article.

Conflicts of Interest
e authors declare no conflicts of interest.

Authors' Contributions
All the authors contributed equally to this study. M. Naeem and A. Q. Baig completed all the computations, wrote the manuscript, and added all figures. X. Zhang checked and corrected the initial manuscript. M. A. Zahid added some final remarks and improved the overall paper. All authors read and approved the final draft.