#### Abstract

Topological indices are very useful to assume certain physiochemical properties of the chemical compound. A molecular descriptor which changes the molecular structures into certain real numbers is said to be a topological index. In chemical graph theory, to create quantitative structure activity relationships in which properties of molecule may be linked with their chemical structures relies greatly on topological indices. The benzene molecule is a common chemical shape in chemistry, physics, and nanoscience. This molecule could be very beneficial to synthesize fragrant compounds. The circumcoronene collection of benzenoid is one family that generates from benzene molecules. The purpose of this study is to calculate the topological indices of the double and strong double graphs of the circumcoronene series of benzenoids . In addition, we also present a numerical and graphical comparison of topological indices of the double and strong double graphs of the circumcoronene series of benzenoid .

#### 1. Introduction and Preliminaries

For undetermined notations and terminologies, we refer the readers to read the book [1].

Let be a simple, finite connected graph, where the set of vertices is and the set of edges is . For every vertex , the edge connecting and is denoted by . In graph , the total number of edges that connects to each vertex is known as the degree of vertex. The number of connected vertices to a fixed vertex is known as neighborhood. The degree of a vertex is denoted by , where . Hand-shaking lemma is very productive for calculating the size of a graph .

Lemma 1. *If a graph is having size , then*

In chemical graph theory, topological indices show a significant role in assisting chemists for modeling the molecular structure of chemical compounds and studying their chemical and physical characteristics. In chemistry, discovery of the drugs commonly relies on the topological descriptors. Drugs are characterized as molecular graphs, where graphs considered are simple with no multiple edges and no cycle formation. These topological descriptors provide information of a chemical compound based on the arrangement of its atoms and their bonds. A wide range of topological indices have been studied, and some of the more frequent forms of topological indices include degree-based, distance-based topological indices, and counting-related polynomials. In the topological indices, very famous and the oldest index is the Wiener index .

The Wiener index [2] is defined as follows:where is the distance among vertices and of a graph .

A graph geometric arithmetic index [3] is defined as follows:

A graph atomic bond connectivity index [4] is defined as follows:

A graph forgotten index [5] is defined as follows:

A graph inverse sum indeg index [6] is defined as follows: A graph general inverse sum indeg index [7] is defined as follows:where and are the real numbers.

A graph first multiplicative-Zagreb and second multiplicative-Zagreb indices are defined [8] as follows:

It is also possible to write the first multiplicative-Zagreb index [9] for as follows:

Imran et al. [10–12] studied the edge Mostar index of nanostructures and chemical structures by using graph operations and also computed the eccentric connectivity polynomial of connected graphs and Mostar indices for melem chain nanostructures. For more details about topological indices, we refer the works of Xiong et al. [13], Hong et al. [14], Alaeiyan et al. [15], Ch et al. [16], and Sardar et al. [17].

*Definition 1. *The well-known family of the benzenoid molecular graph is circumcoronene series of benzenoid , where [18]. This family of graph constructed exclusively from benzene on circumference. Certain main members of circumcoronene series of benzenoid are benzene , coronene , circumcoronene , and circumcircumcoronene [19]. Generally, circumcoronene series of benzenoid is shown in Figure 1.

*Definition 2. *In order to make a double graph of a graph , take two copies of the graph and join the nodes in each copy with their neighbors in the other copy [20]. For example, the graph and its double graph are shown in Figure 2. In double graph of circumcoronene series of benzenoid, there are vertices and edges, respectively. In we have vertices of degree 4 and vertices of degree.

*Definition 3. *Consider the two copies of graph , and by joining the closed neighborhoods of one graph’s vertex to the vertex in an adjacent graph, one can obtain the strong double graph of graph [21]. For example, strong double graph of graph is shown in Figure 3.

This study is laid out as follows. We will examine some vertex-based topological indices of double and strong double graphs of circumcoronene series of benzenoid in Sections 2 and 4, respectively. The comparison is given in Sections 3 and 5. In Section 6, we provide final remarks for the whole study.

#### 2. Degree-Based Topological Indices of Double Graph of Circumcoronene Series of Benzenoid Graph

This section contains a calculation of the degree-based indices of the double graph of circumcoronene series of benzenoid

Theorem 1. *Let be the double graph of circumcoronene series of benzenoid graph ; then, the geometric arithmetic index of is*

*Proof. *In the double graph of circumcoronene series of benzenoid, there are vertices and edges, respectively. There are vertices in of degree 4 and of degree 6.

We separate the edges of into the edges of the type, where is an edge. In , we get edge of types and and . A list of their edges is given in Table 1.

By using Table 1 and equation (1), the result that we obtain is

Theorem 2. *Let be the double graph of circumcoronene series of the benzenoid graph ; then, the index of is*

*Proof. *By using Table 1 and equation (4), the result that we obtain is

Theorem 3. *Let be the double graph of circumcoronene series of benzenoid graph ; then, the is*

*Proof. *By using Table 1 and equation (5), the result that we obtain is

Theorem 4. *Let be the double graph of circumcoronene series of the benzenoid graph ; then, the inverse sum indeg index of is*

*Proof. *By using Table 1 and equation (6), the result that we obtain is

Theorem 5. *Let be the double graph of circumcoronene series of the benzenoid graph ; then, the general inverse sum indeg index () of is*

*Proof. *By using Table 1 and equation (7), the result that we obtain iswhere and are the real numbers.

Theorem 6. *Let be the double graph of circumcoronene series of the benzenoid graph ; then, the first multiplicative-Zagreb index of is*

*Proof. *By using Table 1 and equation (10), the result that we obtain is

Theorem 7. *Let be the double graph of circumcoronene series of the benzenoid graph ; then, the second multiplicative-Zagreb index of is*

*Proof. *By using Table 1 and equation (9), the result that we obtain is

#### 3. Comparison

In this section, we present a numerical and graphical comparison of topological indices that included the first multiplicative-Zagreb index , general inverse sum indeg index (), atom bond connectivity index (), forgotten index (), geometric arithmetic index , second multiplicative-Zagreb index , and inverse sum indeg index for *m* = 1, 2, 3, 4, …, 10 for the double graph of circumcoronene series of the benzenoid graph , as given in Table 2 and Figure 4.

#### 4. Degree-Based Topological Indices of Strong Double Graphs of Circumcoronene Series of Benzenoid Graph

This section contains a calculation of the degree-based indices of the strong double graph of circumcoronene series of benzenoid . Figure 3 shows the strong double graph of .

Theorem 8. *Let be the double graph of circumcoronene series of the benzenoid graph ; then, the geometric arithmetic index of is*

*Proof. *In the strong double graph of circumcoronene series of benzenoid, there are vertices and edges, respectively. There are vertices in of degree 5 and of degree 7.

We separate the edges of into the edges of the type, where is an edge. In , we get edge of types and and . A list of their edges is given in Table 3.

By using Table 3 and equation (1), the result that we obtain is

Theorem 9. *Let be the strong double graph of circumcoronene series of the benzenoid graph ; then, the index of is *

*Proof. *By using Table 3 and equation (4), the result that we obtain is

Theorem 10. *Let be the strong double graph of circumcoronene series of the benzenoid graph ; then, the is*

*Proof. *By using Table 3 and equation (5), the result that we obtain is

Theorem 11. *Let be the strong double graph of circumcoronene series of the benzenoid graph ; then, the inverse sum indeg index of is*

*Proof. *By using Table 3 and equation (6), the result that we obtain is

Theorem 12. *Let be the strong double graph of circumcoronene series of the benzenoid graph ; then, the general inverse sum indeg index () of is*

*Proof. *By using Table 3 and equation (7), the result that we obtain iswhere and are the real numbers.

Theorem 13. *Let be the strong double graph of circumcoronene series of the benzenoid graph ; then, the first multiplicative-Zagreb index of is*

*Proof. *By using Table 3 and equation (10), the result that we obtain is

Theorem 14. *Let be the strong double graph of circumcoronene series of the benzenoid graph ; then, the second multiplicative-Zagreb index of is*

*Proof. *By using Table 3 and equation (9), the result that we obtain is

#### 5. Comparison

In this section, we present a numerical and graphical comparison of topological indices that included the first multiplicative-Zagreb index , general inverse sum indeg index (), atom bond connectivity index (), forgotten index (), geometric arithmetic index , second multiplicative-Zagreb index , and inverse sum indeg index for *m* = 1, 2, 3, 4, …, 10 for the strong double graph of circumcoronene series of the benzenoid graph , as given in Table 4 and Figure 5.

#### 6. Conclusion

We have computed the closed formulae of topological indices such as the first multiplicative-Zagreb index , general inverse sum indeg index (), atom bond connectivity index (), forgotten index (), geometric arithmetic index , second multiplicative-Zagreb index , and inverse sum indeg index of double and strong double graphs of circumcoronene series of benzenoid . Chemical compounds can be studied by these indices in order to understand their diverse properties. The geometric structure and comparison of obtained results are shown graphically and numerically. Those results are convenient for further study as they do not include any polynomial.

#### Data Availability

The data used to support the findings of this study are available from the corresponding author upon request.

#### Conflicts of Interest

The authors declare that they have no conflicts of interest.