On Topological Properties of Symmetric Chemical Structures

: The utilizations of graph theory in chemistry and in the study of molecule structures are more than someone’s expectations, and, lately, it has increased exponentially. In molecular graphs, atoms are denoted by vertices and bonds by edges. In this paper, we focus on the molecular graph of (2D) silicon-carbon Si 2 C 3 - I and Si 2 C 3 - II . Moreover, we have computed topological indices, namely general Randi´c Zagreb types indices, geometric arithmetic index, atom–bond connectivity index, fourth atom–bond connectivity and ﬁfth geometric arithmetic index of Si 2 C 3 - I and Si 2 C 3 - II .


Introduction
Every structural formula that includes covalent bonded compounds or molecules are graphs. Along these lines, they are called molecular graphs. Recently, chemical graph theory tools are used for numeration, systematization of the problem in hand, it provides the process of arranging laws or rules according to a system or planning, nomenclature, it provides the connection between the compounds or atoms, and computer programming. The significance of graph theory tools for science stands for the most part from the presence of isomerism, which is excused by substance diagram hypothesis. The pith of chemistry is the combinatorics of molecules as per clear principles. Along these lines, the most satisfactory numerical apparatuses for this design are graph hypothesis and combinatorics, the branches of arithmetic that are nearly related.
Silicon has numerous preferences over other semiconductor materials: It is, of minimal effort, it is nontoxic, essentially its accessibility is boundless, and behind it many years of involvement in purging, development and gadget creation. It is utilized for all cutting edge electronic gadgets.
The most reliable structures of two-dimensional (2D) silicon-carbon monolayer mixes with different stoichiometric blends were expected in [1] which in light of the molecule swarm streamlining signified as (PSO) method joined with thickness utilitarian hypothesis optimization.
The graphene sheets were effectively confined in [2,3] and from that point forward this honeycomb organized 2D material has roused and enlivened concentrated research interests to a great extent as a result of its surprising mechanical, electronic and optical properties, including its anomalous quantum Lobby influence, overwhelming electronic conductivity, and high mechanical quality. In particular, the intriguing electronic properties of graphene pull in see for this 2D material as a potential probability for applications in speedier and smaller electronic gadgets.
Like carbon, silicon additionally has a 2D allotrope with a honeycomb structure, in particular silicene. To date, bunches of exertion have been given to open a bandgap in silicene sheets. In addition, 2D silicon-carbon (Si − C) monolayers can be seen as creation tunable materials between the unadulterated 2D carbon monolayer-graphene and the unadulterated 2D silicon monolayer-silicene. Loads of endeavors have been directed towards anticipating the most stable structures of the SiC sheet read this [4,5] for more data.
Given a graph G = (V, E) where V is the vertex set and E is the edge set of G, the degree deg(s) of s is the quantity of spokes in G episode with s and is indicated as d(s). A graph can be spoken to by a polynomial, a numerical esteem or by lattice frame. All the concepts of graph theory and combinatorics are used from the book of Harris et al. [6]. There are sure kinds of topological records for the most part capricious based, degree based and remove based files and so forth.
Graovoc et al. [26] introduced the fifth version of geometric arithmetic index GA 5 as:

Applications of Topological Indices
The Randic index is a topological descriptor that has correlated with a lot of chemical characteristics of the molecules and has been found to the parallel to computing the boiling point and Kovats constants of the molecules. The atom-bond connectivity (ABC) index provides a very good correlation for the stability of linear alkanes as well as the branched alkanes and for computing the strain energy of cyclo alkanes [14]. To correlate with certain physico-chemical properties, GA index has much better predictive power than the predictive power of the Randic connectivity index [27]. The first and second Zagreb index were found to occur for computation of the total π-electron energy of the molecules within specific approximate expressions [28]. These are among the graph invariants, who were proposed for measurement of skeleton of branching of the carbon-atom [17].

Methods
To compute our results, we use the method of combinatorial computing, vertex partition method, edge partition method, graph theoretical tools, analytic techniques, degree counting method and sum of degrees of neighbours method. Moreover, we use the Matlab for mathematical calculations and verifications. We also use the maple for plotting these mathematical results.

Silicon Carbide Si 2 C 3 -I[p, q] 2D Structure
The 2D molecular graph of Silicon Carbide Si 2 C 3 -I is given in Figure 1. To describe its molecular graph, we have used the settings in this way: we define p as the number of connected unit cells in a row (chain) and by q we represents the number of connected rows each with p number of cell. In Figure 2, we gave a demonstration how the cells connect in a row (chain) and how one row connects to another row. We will denote this molecular graph by Si 2 C 3 -I[p, q]. Thus, the quantity of vertices in this graph is 10pq and the number of edges are 15pq − 2p − 3q.

Methodology of Silicon Carbide Si 2 C 3 -I[p, q] Formulas
For the computation of these formulas for Silicon Carbide Si 2 C 3 -I[p, q], we use first a unit cell and then combine with another unit cell in horizontal direction and so on up to p unit cells. After this, we use first a unit cell and then combine with another unit cell in the vertical direction and so on up to q unit cells. Thus, we obtained Silicon Carbide [p, q] structure (see Figure 1). Now, for the computation of vertices, we use the Table 1 and Matlab software for generalizing these formulas of vertices. In the following table, V 1 represents the quantity of vertices of degree 1, V 2 represents the quantity of vertices of degree 2 and V 3 represents the quantity of vertices of degree 3. Thus, finally, we calculate he number of vertices of degree 1 are 2, the quantity of vertices of degree 2 are 4p + 2 + 6(q − 1) and the number of vertices of degree 3 are 10pq − 4p − 6q + 2.
To find the abstracted indices, we will partition the edges of Si 2 C 3 -I[p, q] using the above methodology. Moreover, we use the combinatorial counting and standard edge partition.  Table 2 shows the edge partition of Si 2 C 3 -I[p, q] with p, q ≥ 1. Finally, for comparison of these indices, we plot its diagram in maple software (see Figures 3,4,7 and 8).

(d(s), d(t))
Frequency In this section, we compute the general result of topological indices for Si 2 C 3 − I[p, q]. In addition, we construct the tables for these indices for small values of p, q. Moreover, we give graphical comparison and application of these indices.
Let G be the graph of Si 2 C 3 -I[p, q]. Then, from the edge partition of Si 2 C 3 -I[p, q], which is given in Table 2, the atom-bond connectivity index is computed as (see Table 3): After some easy calculations, we get: Let G be the graph of Si 2 C 3 -I[p, q]. Now, by using Table 2, the general Randić index for α = 1 (see Table 4). For α = −1 (see Table 5), , For α = 1 2 (see Table 6), For α = − 1 2 (see Table 7), , Let G be the graph of Silicon Carbide Si 2 C 3 -I[p, q]. Now, by using Table 2, the geometric arithmetic index is computed as below (see Table 8): ,

• First and second Zagreb index
Let G be the graph of Si 2 C 3 -I[p, q]. Now, by using Table 2, the first and second Zagreb indices are computed as below (see Table 9 and 10):  In this section, we presented the comparison of above calculated topological indices for Si 2 C 3 -I[p, q] with p = 1, 2, 3, ..., 1500 and q = 1, 2, 3, ..., 1500 graphically in Figure 3.  Table 11 demonstrates the edge parcel in light of the degree total of end vertices of each edge of the chemical graph Si 2 C 3 -I[p, q] for p, q ≥ 2. (d(s), d(t)) (S u , S v ) Frequency • The fourth atom-bond connectivity index ABC 4 Si 2 C 3 − I[p, q] .
Let G be the graph of Silicon Carbide of type Si 2 C 3 -I[p, q]. Now, by using Table 11, the fourth atom-bond connectivity index is computed as below (see Table 12): After an easy calculation, we get:  Let G be the graph of Si 2 C 3 -I[p, q]. Now, by using Table 11, the fifth geometric arithmetic index is computed as below (see Table 13):  In this section, we presented the comparison of above calculated topological indices for Si 2 C 3 -I[p, q] graphically in Figure 4.

Silicon Carbide Si 2 C 3 -I I[p, q] 2D Structure
The 2D molecular graph of Silicon Carbide Si 2 C 3 -I I is given in Figure 5. To describe its molecular graph, we have used the settings in this way: we define p as the number of connected unit cells in a row (chain) and, by q, we represent the number of connected rows, each with p number of cells.
In Figure 6, we gave a demonstration of how the cells connect in a row (chain) and how one row connects to another row. We will denote this molecular graph by Si 2 C 3 -I I[p, q]. Thus, the quantity of vertices in this graph is 10pq and the number of edges are 15pq − 3p − 3q.
To find the abstracted indices, we will partition the edges of Si 2 C 3 -I I[p, q] using the above methodology. Moreover, we use the combinatorial counting and standard edge partition.  Table 15 shows the edge partition of Si 2 C 3 -I I[p, q] with p, q ≥ 1. (d(s), d(t)) Frequency

Main Results for Silicon Carbide Si 2 C 3 − I I[p, q]
In this section, we compute the general result of topological indices for Si 2 C 3 − I I[p, q]. In addition, we construct the tables for these indices for small values of p, q. Moreover, we give graphical comparison and application of these indices.

• Atom-bond connectivity index ABC Si 2 C 3 -IO[p, q]
Let G be the graph of Silicon Carbide of type Si 2 C 3 -I I[p, q]. Then, from the edge partition of Si 2 C 3 -I I[p, q] which is given in Table 15, the atom-bond connectivity index can be calculated as (see Table 16): Let G be the graph of Si 2 C 3 -I I[p, q]. Now, by using Table 15, the general Randić index for α = 1 (see Table 17), For α = −1 (see Table 18), , For α = 1 2 (see Table 19), For α = −1 2 (see Table 20), , Let G be the graph of Silicon Carbide Si 2 C 3 -I I[p, q]. Now, by using Table 15, the geometric arithmetic index is computed as below (see Table 21): ,

• The first and second Zagreb index
Let G be the graph of Si 2 C 3 -I I[p, q]. Now, by using Table 15, the first Zagreb index is computed as below (see Table 22): The second Zagreb index is computed below (see Table 23): In this section, we presented the comparison of above calculated topological indices for Si 2 C 3 − I I[p, q] graphically in Figure 7.  Table 24 demonstrates the edge parcel in light of the degree total of end vertices of each edge of the chemical graph Si 2 C 3 -I I[p, q] for p, q ≥ 2. (d(s), d(t)) (S u , S v ) Frequency Let G be the graph of Silicon Carbide of type Si 2 C 3 -I I[p, q]. Now, by using Table 24, the fourth atom-bond connectivity index is computed as (see Table 25): After an easy calculation, we get:   We presented the comparison of topological indices for Si 2 C 3 -I I[p, q] graphically in Figure 8.

Comparisons and Discussion
In this section, we have computed all indices for different values of p, q for both structures Si 2 C 3 -I[p, q] and Si 2 C 3 -I I[p, q]. In addition, we construct Tables 3-10, 12 and 13 for small values of p, q for these topological indices to the structure Si 2 C 3 -I[p, q] and Tables 16-23, 25 and 26 to the structure Si 2 C 3 -I I[p, q]. Now, from Tables 27 and 28, we can easily see that all indices are in increasing order as the values of p, q are increases. In addition, on the other hand, indices showed higher values for Si 2 C 3 -I[p, q], as compared to those of Si 2 C 3 -I I[p, q].
The graphical representations of topological indices of Si 2 C 3 -I[p, q] and Si 2 C 3 -I I[p, q] are depicted in Figures 3, 4, 7 and 8 for certain values of p, q. Now, we presented the comparison of all topological indices using Table 27, for Si 2 C 3 -I[p, q] in Figure 9 and using Table 28, for Si 2 C 3 -I I[p, q] in Figure 10.