On Generalized Topological Indices of Silicon-Carbon

Department of General Education, Anhui Xinhua University, Hefei 230088, China School of Mathematics and Physics, Anhui Jianzhu University, Hefei 230601, China Department of Mathematics, Azarbaijan Shahid Madani University, Tabriz, Iran Department of Mathematics, COMSATS University Islamabad, Lahore Campus, Islamabad, Pakistan Department of Mathematics and Applications, Shahed University, Tehran, Iran Faculty of Ocean Engineering Technology and Informatics, University Malaysia Terengganu, Kuala Nerus 21030 UMT, Terengganu, Malaysia


Introduction
Mathematical chemistry is a branch of theoretical chemistry that discusses about the molecular structure by mathematical methods without necessarily referring to quantum mechanics. Molecular descriptors play a significant role in mathematical chemistry especially in QSPR/QSAR investigations [1]. A chemical structure can be represented by using graph theory, where vertices denote atoms and edges denote chemical bonds. Chemical graph theory is a branch of mathematical chemistry that it is a subject that connects mathematics, chemistry, and graph theory and solves problems arising in chemistry mathematically (for more details you can see [2][3][4][5][6][7][8]).
In chemical graph theory, a graph of molecule is a simple connected graph, in which atoms and chemical bonds are represented by vertices and edges, respectively. A graph is connected if there is a connection between any pair of vertices. Among them, special place is reserved for so-called topological descriptors or topological indices. Actually, topological indices are numeric quantities that tell us about the whole structure of graph. e topological indices are useful in the prediction of physicochemical properties and the bioactivity of the chemical compounds [9][10][11]. e topological indices of 2-dimensional silicon-carbons are computed in [12], in [13], Kwunet al. On the Multiplicative Degree-Based Topological Indices of Silicon-Carbon Si 2 C 3 − I[p, q] and Si 2 C3 − II [p, q], in [14], Imran et al.. On Topological Properties of Symmetric Chemical Structures in [15], Idrees et al. Molecular Descriptors of Benzenoid System, in [16], Kulli. F-indices of Chemical Networks, in [6], in [17], Gao et al. the Redefined first, second and third Zagreb Indices of Titania Nanotubes TiO 2 [m, n] and in [18], Kang et al. computed the topological indices of 2-dimensional silicon-carbon. For more details, see [19][20][21][22][23].
Bearing this in mind, it seems to be purposeful to compute the generalization of Zagreb index, the generalized Zagreb index, the first and second hyper F-indices, the sum connectivity F-index, and the product connectivity F-index of Si 2 C 3 − I[p, q], Si 2 C 3 − II [p, q], Si 2 C 3 − III [p, q], and SiC 3 − III [p, q]. roughout this paper, all graphs will be assumed simple that is without loops, multiple, or directed edges. Let G � (n, m) be a simple graph with vertex set V(G) � v 1 , v 2 , v 3 , . . . , v n and edge set E(G), |E(G)| � m. Also, let d i be the degree of vertex v i in graph G, for i � 1, 2, . . . , n. e concept of valence in chemistry and the concept of degree in a graph are somehow closely related. For details on bases of graph theory, we refer to the book [24]. If two vertices u and v of the graph G are adjacent, then the edge connecting them will be denoted by uv. e number of first neighbors of the vertex u ∈ V(G) is its degree and will be denoted by Γ(u).
In [25], the authors defined a new index, named generalization of Zagreb index: where α and β are arbitrary real numbers. Few years later, the same index was proposed in [26] under the name second Gourava index, obtained as a special case of the generalized Zagreb index M r,s introduced in [27]: (2) In [28], Ghobadi et al. defined the hyper F-index or the first hyper F-index of a graph G as (3) In [16], the second hyper F-index of a graph is defined as In [16], the author introduced the sum connectivity F-index and the product connectivity F-index of a graph G, defined as e concept of silicon carbide was introduced by an American scientist in 1891. But nowadays, we can produce silicon carbide artificially by silica and carbon. Till 1929, silicon carbide was known as the hardest material on Earth.
Here, we will find out the generalization of Zagreb index, the generalized Zagreb index, the first and second hyper F-indices, the sum connectivity F-index, and the product connectivity F-index. is paper is organized as follows. In Section 2, we compute the generalization of Zagreb index, the generalized Zagreb index, the first and second hyper F-indices, the sum connectivity F-index, and the product connectivity F-index graphs of Si 2 C 3 − I[p, q]. In Section 3, we compute the generalization of Zagreb index, the generalized Zagreb index, the first and second hyper F-indices, the sum connectivity F-index, and the product connectivity F-index graphs of Si 2 C 3 − II [p, q]. In Section 4, we compute the generalization of Zagreb index, the generalized Zagreb index, the first and second hyper F-indices, the sum connectivity F-index, and the product connectivity F-index graphs of Si 2 C 3 − III [p, q]. In Section 5, we compute the generalization of Zagreb index, the generalized Zagreb index, the first and second hyper F-indices, the sum connectivity F-index, and the product connectivity F-index graphs of SiC 3 − III [p, q].

Results for Silicon-Carbon
In this section, we compute the generalization of Zagreb index, the generalized Zagreb index, the first and second hyper F-indices, the sum connectivity F-index, and the product connectivity F-index graphs of Si 2 C 3 − I[p, q]. In Figure 1, one unit of Si 2 C 3 − I is shown. Molecular graph of S i 2C 3 − I is shown in Figure 2, in which p denotes the number of cells attached in a single row and q denotes the number of total rows where each row contains p cells. In Figures 3 and 4, we demonstrate how cells are connected in one row (chain) and how one row is connected to another row. In Figures 1-4, carbon atoms are shown as brown, and silicon atoms Si are shown as blue.
We start by proving the carbon nanocones for the redefined Zagreb indices.
Journal of Mathematics . e edge set of the Si 2 C 3 − I[p, q] can be partitioned as follows: From the molecular graph of Si 2 C 3 − I[p, q], we can observe that |E 1 | � 1, us, by definition generalization Zagreb index of Si 2 C 3 − I[p, q], we have which is the required (18) result. By definition of the generalized Zagreb index of which is the required (10) result. By definition of the second hyper F-index of PF Proof. Consider the graph silicon carbide Si 2 C 3 − I[p, q]. By Remark 1, the graph Si 2 C 3 − I[p, q] contains 10pq vertices and 15pq − 2p − 3q edges. By definition of the sum con- which is the required (14) result. By definition of product connectivity F-index of Journal of Mathematics which is the required (15) result.

Results for Silicon-Carbon
In this section, we compute the generalization of Zagreb index, the generalized Zagreb index, the first and second hyper F-indices, the sum connectivity F-index, and the product connectivity F-index graphs of Si 2 C 3 − II[p, q].In Figure 5, one unit of Si 2 C 3 − II is given. By connecting p cells in a row and then connecting q rows where each row contains p cells, we get molecular graph of Si 2 C 3 − II. e molecular graph of Si 2 C 3 − II is shown in Figure 6 for p � 3 and q � 4. Figures 7 and 8 demonstrate how cells are connected in a row (chain) and how a row is connected to another row. We will use Si 2 C 3 − II[p, q] to represent this molecular graph.
We start by proving the silicon carbide for the generalization of Zagreb index.
can be partitioned as follows: Journal of Mathematics 7 From the molecular graph of us, by definition generalization of which is the required (18) result.

By definition of the second hyper F-index of
which is the required (24) result. □ Theorem 6. Let Si 2 C 3 − II[p, q] be the silicon carbide. en, 10 Journal of Mathematics which is the required (27) result.

By definition of product connectivity F-index of
which is the required (28) result.

Results for Silicon-Carbon Si 2 C 3 − III[p, q]
In this section, we compute the generalization of Zagreb index, the generalized Zagreb index, the first and second hyper F-indices, the sum connectivity F-index, and the product connectivity F-index graphs of Si 2 C 3 − III[p, q]. e 2D silicon-carbon (Si − C) single layers can be seen as configurable (or tunable) materials between the pure 2D carbon single-layer graphene and the pure 2D silicon singlelayer silicene. Lots of attempts have been conducted trying anticipating the most stable structures of the SiC sheet (for more details, see [40,41]). e 2D molecular graph of silicon carbide Si 2 C 3 − III[p, q] is given in Figure 9, where carbon atom C is shown in brown color and silicon atom Si is shown in blue color (for more details, see [42]). In Figure 10, we gave a demonstration how the cells connect in a row (chain) and how one row connects to another row; red lines (edges) show the connection between the unit cell in a chain and green lines (edges) connect the upper and lower rows (chains). We will denote this molecular graph by Remark 3 (see [13]). e graph Si 2 C 3 − I[p, q] contains 10pq vertices and 15pq − 2p − 3q edges.
We start by proving the carbon nanocones for the redefined Zagreb indices.  [p, q] can be partitioned as follows:

Theorem 7. Let Si 2 C 3 − III[p, q] be the silicon carbide. en,
which is the required (31) result. By definition of the generalized Zagreb index of which is the required (36) result. By definition of the second hyper F-index of Si 2 C 3 − III[p, q], we have which is the required (37) result. □ Theorem 9. Let Si 2 C 3 − III[p, q] be the silicon carbide. en, which is the required (40) result. By definition of product connectivity F-index of Journal of Mathematics 15 which is the required (41) result.

Results for Silicon-Carbon Si 2 C 3 − III[p, q]
In this section, we compute the generalization of Zagreb index, the generalized Zagreb index, the first and second hyper F-indices, the sum connectivity F-index, and the product connectivity F-index graphs of Si 2 C 3 − III [p, q]. e 2D molecular graph of silicon carbide SiC 3 − III is given in Figure 11, where carbon atom C is shown in brown color and silicon atom Si is shown in blue color (for more details, see [42]). In Figure 12, we gave a demonstration how the cells connect in a row (chain) and how one row connects to another row; red lines show the connection between the unit cells and green lines (edges) connect the upper and lower rows. We will denote this molecular graph by Si 2 C 3 − III [p, q].
We start by proving the carbon nanocones for the redefined Zagreb indices.
can be partitioned as follows: which is the required (54) result.

Conclusion
In this paper, we computed the generalization of Zagreb index, the generalized Zagreb index, the first and second hyper F-indices, the sum connectivity F-index, and the product connectivity F-index graphs of Data Availability e data used to support the findings of this study are cited at relevant places within the text as references.

Conflicts of Interest
e authors declare that there no conflicts of interest.