Document Type : Original Research Article


1 School of Mathematics, Minhaj University, Lahore, Pakistan

2 Minhaj University Lahore, Pakistan

3 Faculty of Education, Van Yuzuncu Yıl University, Van, Turkey


Silicon is a semiconductor material with several advantages over the other similar materials, such as its low cost, nontoxicity, and almost limitless availability, as well as many years of expertise in its purification, manufacture, and device development. It is used in practical for all of the most recent electrical products. Graph theory can be used to depict a chemical structure, with vertices representing atoms and edges representing chemical bonds. Molecular descriptors are important in mathematical chemistry, particularly in QSPR/QSAR research. In this research, by using two graph operations, namely; double and strong double graph, we computed the closed formulas for some degree-based topological indices of silicon carbide. Furthermore, we also compare topological indices numerically and graphically.

Graphical Abstract

On topological indices of double and strong double graph of silicon carbide Si2C3-I[p,q]



In theoretical chemistry (mostly in QSAR/QSPR research), environmental chemistry, pharmacology, and toxicology, topological indices have numerous applications. Let G be an undirected simple graph with  V (G)={r1, r2,r3,... rn} and  E (G)= {e1, e2, e3, ..., en} as the vertex and edge sets. The degree of a vertex  is the number of edges incident to r , and it is symbolized by . For undetermined terminologies and notations, we mention Robin J. Wilson book [1].

Mathematical chemistry provides useful tools like polynomials and functions that rely on information contained in the symmetry of graphs of chemical compounds and very helpful for the prediction of the understudy molecular compound and its characteristics without the usage of quantum mechanics. A topological index is a numerical parameter that describes the topology of a graph. It quantitatively describes the structure of molecules and is utilized in the establishment of qualitative structure activity relationships. Numerous topological indices existing [2,3], but here we investigate the topological indices which based on the degree of vertex [4-6]. Chemical graph theory is a field of mathematical chemistry in which we use graph theory methods to represent chemical activities mathematically.

Topological indices that will be explored in this article are given in Equations (1-7).

In topological indices, Geometric-Arithmetic index is associated with Physico-chemical properties such as entropy and enthalpy of vaporization. Consider a graph G, then geometric-arithmetic index (GA) is defined [7] as:

The atom–bond connectivity index (ABC) is a degree-based graph invariant. It can be used to simulate the thermodynamic properties of organic chemical compounds. Consider a graph G, then atom bond connectivity index (ABC) is defined [8] as follows:

The well-known index is forgotten topological index  (F) which is very helpful for medical scientists to grasp chemical characteristics of the new drugs and is defined [9] as follows:

The Inverse sum index  (ISI) is defined [10] as:

The First and the second multiplicative-Zagreb index multiplicative-Zagreb is used to examine the extreme molecular graphs and is defined [11,12] as follows:

We can also write first multiplicative-Zagreb index [13] as:

For more comprehensive and detailed study on indices and graphs, we mention the following articles [14-19, 23-48] for readers.

Silicon Carbide Si2C3-I[p,q] is a molecular graph in two dimensions [20], depicted in Figure 11. We used the following parameters to characterize its molecular graph: We define p as the number of linked unit cells in a row (chain) as in Figure 1(a), and  q is the number of linked rows, each having p cells. In Figure 1(b), we showed how the cells in a row (chain) link to each another and how one row relates to another row. Figure 1(c) displays the structure of one-dimensional unit cell.

The double graph of silicon carbide is symbolized by D[Si2C3-I(p,q)].Assume two copies of a Si2C3-I(p,q), where (p,q=1) and join each vertex in one copy to its neighbor in the other copy to produce the double graph [21] of the silicon carbide Si2C3-I(p,q) . For example, the double graph of silicon carbide  Si2C3-I(1,1) is depicted in Figure 2. While for strong double graph SD[G] of the silicon carbide, Si2C3-I(p,q) is attained by taking two graphs of Si2C3-I(p,q), where (p,q=1) and joining the closed neighborhoods of each vertex in one graph to the adjacent vertex in the other graph [22]. The SD[Si2C3-I(1,1)] is depicted in Figure 4.

Main results for double graph of silicon carbide

Here, we calculate the indices which base on the degree of vertices, for the double graph of the silicon carbide Si2C3-I(p,q) . In  D[Si2C3-I(p,q)], we have vertices of degree , and . Edges of  D[Si2C3-I(p,q)] are split into the edges of type E[dr,ds], where  is represent the edge. Silicon carbide Si2C3-I(p,q) contains the edges of the type E(2,4), E(2,6), E(4,4), E(4,6), and E(6,6). Table 1 represents the edges of these types.

Theorem 2.1. Let D[Si2C3-I(p,q)] be the double graph of silicon carbide Si2C3-I(p,q). Then,

Proof: Consider the double molecular graph of silicon carbide D(Si2C3-I(p,q)), contains 20pq vertices and 60pq-8p-121 edges. By using Equation (1) and Table 1, the  GA index calculated as follows:

Via Equation (2) and Table 1, the  index is calculated as:


In this section, we compute a numerical and graphical comparison of topological indices based on the degree of the vertex, which are computed above for the double graph of silicon carbide [D(Si2C3-I[p,q])], where  p = 1, 2, 3, . . . , 10 and q = 1, 2, 3, . . . , 10.

Main results for strong double graph of silicon carbide (Si2C3-I(p,q))  

We will compute indices which are based on the degree of vertices, for the strong double graph of the silicon carbide in this section. In  (Si2C3-I(p,q)), we have vertices of degree 3, 5, and 7. Edges of  SD(Si2C3-I(p,q)) were split into the edges of type E[dr, ds], where rs is represent the edge. Silicon carbide Si2C3-I(p,q) contains edges of the type E(2,2), E(3,5), E(3,7), E(5,5), E(5,7), and E(7,7). Table 3 represents the edges of these types.


The findings of this study can help to understand the physical features and biological activities of silicon carbide. In this paper, we investigated the topological indices namely; Inverse sum indeg index , the first multiplicative-Zagreb index , atom bond connectivity index ( ), forgotten index ( ), geometric arithmetic index , the second multiplicative-Zagreb index  of strong double and double graph of silicon carbide  The comparison and geometric structure of attained results are presented numerically and graphically.


The authors would like to thank the reviewers for their helpful suggestions and comments.

Conflict of Interest

The authors declare that there is no conflict of interests regarding the publication of this manuscript.


Muhammad Shoaib Sardar:

Muhammad Asad Ali:

Murat Cancan:




Copyright © 2023 by SPC (Sami Publishing Company) + is an open access article distributed under the Creative Commons Attribution License(CC BY)  license  (, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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