Review Article  Open Access
Muhammad Asad Ali, Muhammad Shoaib Sardar, Imran Siddique, Dalal Alrowaili, "VertexBased Topological Indices of Double and Strong Double Graph of Dutch Windmill Graph", Journal of Chemistry, vol. 2021, Article ID 7057412, 12 pages, 2021. https://doi.org/10.1155/2021/7057412
VertexBased Topological Indices of Double and Strong Double Graph of Dutch Windmill Graph
Abstract
A measurement of the molecular topology of graphs is known as a topological index, and several physical and chemical properties such as heat formation, boiling point, vaporization, enthalpy, and entropy are used to characterize them. Graph theory is useful in evaluating the relationship between various topological indices of some graphs derived by applying certain graph operations. Graph operations play an important role in many applications of graph theory because many big graphs can be obtained from small graphs. Here, we discuss two graph operations, i.e., double graph and strong double graph. In this article, we will compute the topological indices such as geometric arithmetic index , atom bond connectivity index , forgotten index , inverse sum indeg index , general inverse sum indeg index , first multiplicativeZagreb index and second multiplicativeZagreb index , fifth geometric arithmetic index , fourth atom bond connectivity index of double graph, and strong double graph of Dutch Windmill graph .
1. Introduction and Preliminaries
For undetermined notations and terminologies, we recommend Robin J. Wilson book [1].
Assume that is a simple graph that has no multiple edges and loops. and are the vertex and edge sets of graph , respectively. The number of elements in and represents the order and size of graph . Vertex degree is the number of edges joining to a vertex in a graph . A vertex degree is indicated by and , where . The following lemma is useful for computing the total number of edges in a graph .
Lemma 1. If is a graph of size , then
This is also called the handshake lemma and was observed by Lenford Euler in 1736. This observation is often called the first theorem of graph theory [2].
The chemical graph theory connects graph theory and chemistry to solve organic chemistry problems. Structuredproperty (QSPR) and structuredactivity (QSAR) relationships are among the most important topics in this field. QSPR/QSAR research relies heavily on topological indices. These topological indices analyse the structure of any finite graph and are based on mathematical equations. Several different kinds of topological indices exist, i.e., degreebased topological indices [3–5], distancebased topological indices [6], and countingrelated topological indices [7, 8]. The topological index concept comes from the work of Wiener, who introduced the Wiener index, and thus, topological indexing history begins.
The Wiener index is defined in [9] as follows:where is the order pair of vertices in and is the distance of vertex rs in .
The geometricarithmetic index [10] of graph is defined as
The atomic bond connectivity index of graph is defined [11] as
The forgotten index is defined [12] as
The inverse sum indeg index is defined [13] as
The general inverse sum indeg index is defined [14] aswhere and are some real numbers.
The first multiplicativeZagreb () and second multiplicativeZagreb index index is defined [15] as
The first multiplicativeZagreb index can also be written in the sum of the edges [16] of :
The index is defined [17] aswhere is summation of degrees of all neighbor of vertex and the same for .
The index is defined [18] as
We suggest that readers read the following articles for more detailed information on topological indices and molecular graphs [19–22].
Definition 1. A Dutch Windmill graph [23] is the graph which is formed by taking the copies of the cycle graph by taking one mutual vertex. It is denoted by where . The order of Dutch Windmill graph is and size . Dutch Windmill graph is depicted in Figure 1.
Definition 2. The double graph of graph is represented by . Assume two copies of a graph, and join each vertex in one copy to its neighbor in the other copy in order to produce the double graph of the graph [24]. For example, the double graph of Dutch Windmill graph is depicted in Figure 2.
(a)
(b)
Definition 3. The strong double graph of the graph is attained by taking two graphs and joining the closed neighbourhoods of each vertex in one graph to the adjacent vertex in the other graph [25]. For example, strong double graph of graph is depicted in Figure 3. A new type of equienergetic and Lequienergetic graph has been found by using strong double graphs.
The following is the structure of this paper. Sections 2 and 3 will analyse some degreebased topological indices of double graphs and strong double graphs of Dutch Windmill graphs, respectively. In Section 4, we provide concluding remarks for the entire paper.
2. DegreeBased Topological Indices of Double Graph of Dutch Windmill Graph
We will compute the vertexbased indices of the double graph of the Dutch Windmill graph in this section.
Theorem 4. Let be the double graph of Dutch Windmill graph . Then, the geometric arithmetic index of is
Proof. The total number of vertices and edges in the double graph of Dutch Windmill graph is and 12p, respectively. In , we have 4p vertices of degree 4 and 2 vertices of degree 8p. We spilt the edges of into those of the type in which r, s are edges. contains edges of type and , and Table 1 presents the edges of these types.
By applying equation (3) and Table 1, we acquire the desired results, i.e.,

Theorem 5. Let be the double graph of Dutch Windmill graph . Then, the index of is
Proof. By applying equation (4) and Table 1, we acquire the desired results, i.e.,
Theorem: 6. Let be the double graph of Dutch Windmill graph . Then, the forgotten index of is
Proof. By applying equation (5) and Table 1, we acquire the desired results, i.e.,
Theorem 7. Let be the double graph of Dutch Windmill graph . Then, the inverse sum indeg index of is
Proof. By applying equation (6) and Table 1, we acquire the desired results, i.e.,
Theorem 8. Let be the double graph of Dutch Windmill graph Then, the general inverse sum indeg index of is
Proof. By applying equation (7) and Table 1, we acquire the desired results, i.e.,
Theorem 9. Let be the double graph of Dutch Windmill graph Then, the first multiplicativeZagreb index of is
Proof. By applying equation (9) and Table 1, we acquire the desired results, i.e.,
Theorem 10. Let be the double graph of Dutch Windmill graph . Then, the second multiplicativeZagreb index of is
Proof. By applying equation (9) and Table 1, we acquire the desired results, i.e.,
Theorem 11. Let be the double graph of Dutch Windmill graph Then, index of is
Proof. We spilt the edges of into those of the type in which “” is an edge. contains edge of the type and , and Table 2 presents the edges of these types.
By applying equation (11) and Table 2, we acquire the desired results, i.e.,

Theorem 12. Let be the double graph of Dutch Windmill graph Then, the fourth atom bond connectivity index of is
Proof. By applying equation (12) and Table 2, we acquire the desired results, i.e.,
3. DegreeBased Topological Indices of Strong Double Graphs of Dutch Windmill Graph
We will compute the vertexbased indices of the strong double graph of the Dutch Windmill graph in this section. The strong double graph of is depicted in Figure 4.
(a)
(b)
Theorem 13. Let be the strong double graph of Dutch Windmill graph . Then, the geometric arithmetic index of is
Proof. The total number of vertices and edges in are and , respectively. In , we have vertices of degree 5 and 2 vertices of degree . We spilt the edges of into those of the type in which “” is an edge. contains edges of the type , , and , and Table 3 presents the edges of these types.
By applying Equation (3) and Table 3, we acquire the desired results, i.e.,

Theorem 14. Let be the strong double graph of Dutch Windmill graph . Then, index for
Proof. By applying equation (4) and Table 3, we acquire the desired results, i.e.,
Theorem 15. Let be the strong double graph of Dutch Windmill graph . Then, the forgotten index of is
Proof. By applying equation (5) and Table 3, we acquire the desired results, i.e.,
Theorem 16. Let be the strong double graph of Dutch Windmill graph . Then, the inverse sum indeg index of is
Proof. By applying equation (6) and Table 3, we acquire the desired results, i.e.,
Theorem 17. Let be the strong double graph of Dutch Windmill graph . Then, general inverse sum indeg index of is
Proof. By applying equation (7) and Table 3, we acquire the desired results, i.e.,
Theorem 18. Let be the strong double graph of Dutch Windmill graph . Then, the first multiplicativeZagreb index of is
Proof. By applying equation (10) and Table 3, we acquire the desired results, i.e.,
Theorem 19. Let be the strong double graph of Dutch Windmill graph . Then, the second multiplicativeZagreb index of is
Proof. By applying equation (9) and Table 3, we acquire the desired results, i.e.,
Theorem 20. Let be the strong double graph of Dutch Windmill graph . Then, the index of is
Proof. We spilt the edges of into those of the type in which “” is an edge. contains edges of the type , , and , and Table 4 presents the edges of these types.
By applying equation (11) and Table 4, we acquire the desired results, i.e.,

Theorem 21. Let be the strong double graph of Dutch Windmill graph . Then, the fourth atom bond connectivity index of is
Proof. By applying equation (11) and Table 4, we acquire the desired results, i.e.,
4. Conclusion
The topological indices provide key information about a molecule's chemical structure and chemical activity, for example, in order to derive quantitative structureactivity relationships (QSARs). These models are derived by applying statistical measures of the molecular structure or properties with descriptors representative of the biological activity (including undesirable side effects) of chemicals (drugs, toxicants, and environmental pollutants). Drug discovery, lead optimization, and toxicity prediction are just a few of the areas in which QSAR is being used. Our purpose in this article was to construct two new graphs from the Dutch Windmill graph using two graph operations, namely, double graph and strong double graph. After that, we calculated some vertexbased topological indices of double graph and strong double graph of the Dutch Windmill graph. Many graph theory applications rely on graph operations. We recommend the readers to compute the topological indices for double and strong double graphs of some other classes of graphs or networks.
Data Availability
No data were used in this manuscript.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
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Copyright
Copyright © 2021 Muhammad Asad Ali et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.