Computation of Topological Indices of Double and StrongDouble Graphs of Circumcoronene Series of Benzenoid (Hm)

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. *e benzene molecule is a common chemical shape in chemistry, physics, and nanoscience. *is molecule could be very beneficial to synthesize fragrant compounds. *e circumcoronene collection of benzenoidHm is one family that generates from benzenemolecules.*e purpose of this study is to calculate the topological indices of the double and strong double graphs of the circumcoronene series of benzenoids (Hm). 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 (Hm).


Introduction and Preliminaries
For undetermined notations and terminologies, we refer the readers to read the book [1].
Let Ɠ(V, E) be a simple, finite connected graph, where the set of vertices is V(Ɠ) and the set of edges is E(Ɠ). For every vertex x ∈ V(Ɠ), the edge connecting x and z is denoted by xz. In graph Ɠ, the total number of edges that connects to each vertex is known as the degree of vertex. e number of connected vertices to a fixed vertex is known as neighborhood.
e degree of a vertex is denoted by d x , where x ∈ V(Ɠ ). Hand-shaking lemma is very productive for calculating the size of a graph Ɠ.

Lemma 1. If a graph Ɠ is having size k, 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. ese 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 W(Ɠ). e Wiener index [2] is defined as follows: where d(x, z) is the distance among vertices x and z of a graph Ɠ.
A graph Ɠ's geometric arithmetic index (GA) [3] is defined as follows: A graph Ɠ' s atomic bond connectivity index (ABC) [4] is defined as follows: A graph Ɠ's forgotten index (F) [5] is defined as follows: A graph Ɠ 's inverse sum indeg index (ISI) [6] is defined as follows: A graph Ɠ's general inverse sum indeg index (ISI (α,β) ) [7] is defined as follows: where α and β are the real numbers. A graph Ɠ's first multiplicative-Zagreb (PM 1 ) and second multiplicative-Zagreb indices (PM 2 ) are defined [8] as follows: It is also possible to write the first multiplicative-Zagreb index (PM 1 ) [9] for Ɠ as follows: Imran et al. [10][11][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 2.
In order to make a double graph D(H m ) of a graph G, take two copies of the graph G and join the nodes in each copy with their neighbors in the other copy [20]. For example, the graph (H 1 ) and its double graph D(H 1 ) are shown in Figure 2. In double graph of circumcoronene series of benzenoid, there are 12m 2 vertices and 4(9m 2 − 3m) edges, respectively. In D(H m ), we have 12m vertices of degree 4 and 12(m 2 − m) vertices of degree.
Definition 3. Consider the two copies of graph G, 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 SD(G) of graph G [21]. For example, strong double graph of graph H 1 is shown in Figure 3.
is 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 (H m ) in Sections 2 and 4, respectively. e comparison is given in Sections 3 and 5. In Section 6, we provide final remarks for the whole study.

Degree-Based Topological Indices of Double Graph of Circumcoronene Series of Benzenoid Graph (H m )
is section contains a calculation of the degree-based indices of the double graph of circumcoronene series of benzenoid (H m ).

Theorem 1. Let D(H m ) be the double graph of circumcoronene series of benzenoid graph (H m ); then, the geometric arithmetic index of D(H m ) is
Proof. In the double graph of circumcoronene series of benzenoid, there are 12m 2 vertices and 4(9m 2 − 3m) edges, respectively. ere are 12m vertices in D(H m ) of degree 4 and 12(m 2 − m) of degree 6. We separate the edges of D(H m ) into the edges of the typeE[d x , d z ], where xz is an edge. In D(H m ), we get edge of types E (4,4) and E (4,6) and E (6,6) . A list of their edges is given in Table 1.
Proof. By using Table 1 and equation (6), the result that we obtain is

Theorem 5. Let D[H m ] be the double graph of circumcoronene series of the benzenoid graph (H m ); then, the general inverse sum indeg index (ISI (α,β) ) of D(H m ) is
Proof. By using Table 1 and equation (7), the result that we obtain is where α and β are the real numbers.

Theorem 6. Let D[H M ] be the double graph of circumcoronene series of the benzenoid graph (H m ); then, the first multiplicative-Zagreb index of D(H m ) is
Proof. By using Table 1 and equation (10), the result that we obtain is

Theorem 7. Let D[H m ] be the double graph of circumcoronene series of the benzenoid graph (H m ); then, the second multiplicative-Zagreb index of D(H m ) is
Proof. By using Table 1 and equation (9), the result that we obtain is (24) □

Comparison
In this section, we present a numerical and graphical comparison of topological indices that included the first multiplicative-Zagreb index (PM 1 ), general inverse sum indeg index (ISI (α,β) ), atom bond connectivity index (ABC), forgotten index (F), geometric arithmetic index (GA), second multiplicative-Zagreb index (PM 2 ), and inverse sum indeg index (ISI) for m � 1, 2, 3, 4, . . ., 10 for the double graph of circumcoronene series of the benzenoid graph (D(H m )), as given in Table 2 and

Degree-Based Topological Indices of Strong Double Graphs of Circumcoronene Series of Benzenoid Graph (H m )
is section contains a calculation of the degree-based indices of the strong double graph of circumcoronene series of benzenoid (H m ). Figure 3 shows the strong double graph of (H 1 ).

Theorem 8. Let SD(H M ) be the double graph of circumcoronene series of the benzenoid graph (H m ); then, the geometric arithmetic index of SD(H m ) is
Proof. In the strong double graph of circumcoronene series of benzenoid, there are 12m 2 vertices and 6(7m 2 − 2m) edges, respectively. ere are 12m vertices in SD(H m ) of degree 5 and 12m(m 2 − 1) of degree 7.
We separate the edges of SD(H m ) into the edges of the typeE(d x , d z ), where xz is an edge. In SD(H m ), we get edge of types E (5,5) and E (5,7) and E (7,7) . A list of their edges is given in Table 3. By using Table 3 and equation (1), the result that we obtain is GA SD H m � (6m + 24) + 48(m − 1) Proof. By using Table 3 and equation (4), the result that we obtain is    Proof. By using Table 3 and equation (9), the result that we obtain is

Comparison
In this section, we present a numerical and graphical comparison of topological indices that included the first multiplicative-Zagreb index (PM 1 ), general inverse sum indeg index (ISI (α,β) ), atom bond connectivity index (ABC), forgotten index (F), geometric arithmetic index (GA), second multiplicative-Zagreb index (PM 2 ), and inverse sum indeg index (ISI) for m � 1, 2, 3, 4, . . ., 10 for the strong double graph of circumcoronene series of the benzenoid graph (SD(H m )), as given in Table 4 and Figure 5.

Conclusion
We have computed the closed formulae of topological indices such as the first multiplicative-Zagreb index (PM 1 ), general inverse sum indeg index (ISI (α,β) ), atom bond connectivity index (ABC), forgotten index (F), geometric arithmetic index (GA), second multiplicative-Zagreb index (PM 2 ), and inverse sum indeg index (ISI) of double and strong double graphs of circumcoronene series of benzenoid H m (m ≥ 1). Chemical compounds can be studied by these indices in order to understand their diverse properties. e geometric structure and comparison of obtained results are shown graphically and numerically. ose results are convenient for further study as they do not include any polynomial.

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

Conflicts of Interest
e authors declare that they have no conflicts of interest.