On Degree-Based Topological Indices for Strong Double Graphs

A topological index is a characteristic value which represents some structural properties of a chemical graph. We study strong double graphs and their generalization to compute Zagreb indices and Zagreb coindices. We provide their explicit computing formulas along with an algorithm to generate and verify the results. We also find the relation between these indices. A 3D graphical representation and graphs are also presented to understand the dynamics of the aforementioned topological indices.


Introduction
Chemical graph theory is an important topological field of mathematical chemistry that deals with mathematical modelling of chemical compound structures. A molecular structure of a compound consists of many atoms. Specially, hydrocarbons are chemical compounds which consist of carbon and hydrogen atoms. A graph consisting of hydrocarbons is known as a molecular graph which represents the carbon structure of a molecule [1].
We consider a simple molecular graph, say G, which consists of nonhydrogen atoms and covalent bonds. In graph theory, the nonhydrogen atoms are represented by a set of vertices V � V(G) and the covalent bonds with the set of edges E � E(G). e number of atoms and bonds in a structure is represented by n � |V| and m � |E|, respectively. e valency of an atom is represented by R G (v), and it is known as the degree of vertex v ∈ V(G), which also represents the number of adjacent (or neighboring) vertices of

respectively.
A molecular descriptor is known as the topological index provides specific information about the structure of molecules. In graph theory, the molecular structure is considered as a graph G. e topological index is also known as the connectivity index [2,3]. Topological indices are largely applied in chemistry to develop the quantitative structureactivity relationship (QSAR) in which the characteristics of molecules can be correlated with their chemical structures [4]. e physicochemical properties of a molecule can also be explained through topological indices. e first index of a chemical graph was introduced by Harold Wiener [5] in 1947 as an aid to determining the boiling point of the paraffin compound. is index is known as the Wiener index and defined as where the notation d (u, v) represents the distance between u and v.
A topological index is defined as a function T: ψ ⟶ R, where ψ is the set of finite simple graphs and R is a set of real numbers which satisfy the relation T(G) � T(H) if G is isomorphic to H. Recently published work [6,7] motivated us to further investigate the Zagreb indices and coindices of strong double graphs. e first Zagreb index M 1 (G) and second Zagreb index M 2 (G) were introduced by Gutman and Trinajstić in 1972 [8] and elaborated by Nikolić et al. after 30 years in 2003 [9]. M 1 (G) and M 2 (G) are defined as (1) e first Zagreb index is also written as Recently, some useful versions of Zagreb indices have been discovered, such as multiplicative Zagreb indices [3,7,10], multiplicative sum Zagreb indices [11,12], Zagreb coindices [6], and multiplicative Zagreb coindices [13]. e important variants of the Zagreb index are the first and second Zagreb coindices, which are defined as follows, respectively: Doslic [14] introduced M 1 (G) and M 2 (G) in 2008. In 2009, Ashrafi et al. [15] determined the extremal values of these new invariants for some special graphs. ey [6] also explored their fundamental properties and provided some explicit formulas for these versions under different graph operations.

Main Results
In this section, we study Zagreb indices and Zagreb coindices of strong double graphs. We also study these indices for generalized k-iterated strong double graphs. We use the concept of edge partition to reduce computation complexity and obtain computing formulas for these indices.
For the sake of simplicity, we consider S D(G) � G 1 * � G * and G k * � (G (k− 1) * ) * for k ≥ 2. Assume that G 0 * � G for the sake of consistency.
In the following theorem, we study the first and second Zagreb indices of the strong double graph. Theorem 1. Let G be a simple connected graph of order n and size m; then, Proof. For the sake of convenience, we label all vertices in G as v 1 , v 2 , . . . , v n . Suppose that x i and y i are the corresponding clone vertices, in strong double graph G * , of v i for each i � 1, . . . , n.
For any given vertex v i in G and its clone vertices x i and

Journal of Chemistry
For So, we only need to consider the total contribution of the following three types of adjacent vertex pairs both to M 1 (G * ) and to M 2 (G * ).
Type 2: the adjacent vertex pairs x i , y i for each i � 1, . . . , n Type 3: the adjacent vertex pairs x i , y j and y i , e total contribution of adjacent vertex pairs of type 1 in M 1 (G * ) is given by and M 2 (G * ) is given by e total contribution of adjacent vertex pairs of type 2 in M 1 (G * ) is given by and M 2 (G * ) is given by e total contribution of adjacent vertex pairs of type 3 in M 1 (G * ) is given by Journal of Chemistry 3 and M 2 (G * ) is given by erefore, by using equations (3), (5) and (7), we have By using (4), (6) and (8), we also have In this theorem, we study the first and second Zagreb coindices of the strong double graph. □ Theorem 2. Let G be a simple connected graph of order n and size m; then, Proof. For the sake of convenience, we label all vertices in G as v 1 , v 2 , . . . , v n . Suppose that x i and y i are the corresponding clone vertices, in strong double graph G * , of v i for each i � 1, . . . , n.

by the definition of the strong double graph.
For So, we only need to consider the total contribution of the following two types of nonadjacent vertex pairs both to M 1 (G * ) and to M 2 (G * ). Journal of Chemistry and M 2 (G * ) is given by e total contribution of nonadjacent vertex pairs of type 2 in M 1 (G * ) is given by and M 2 (G * ) is given by Journal of Chemistry 5 erefore, by using equations (11) and (13), we have By using equations (12) and (14), we also have Now, we present the first and second indices and coindices of k-iterated strong double graphs. □ Theorem 3. Let G be a nontrivial graph of order n and size m, and let G k * be its kiterated strong double graph. en, Proof. For any nontrivial graph G with n vertices and m edges, the number of vertices in G * is 2n and the number of edges in G * is 2m plus those edges between the sets x 1 , x 2 , . . . , x n and y 1 , y 2 , . . . , y n , that is, 4 k m + n. Now, we deduce that G k * has 2 k m vertices and 4 k m + (2 2k− 1 − 2 k− 1 )n edges.
As we know, Using the size of strong double graph m � 4 k m + (2 2k− 1 − 2 k− 1 )n, we have By eorem 1 and the definition of the k-iterated strong double graph, for k ≥ 1, we have 6 Journal of Chemistry As we know, Input: A is an adjacency matrix of a finite simple connected graph Output: IM 1 , IM 2 , CM 1 , CM 2 Variables used: nr � number of rows in A, nc � number of columns in A, (1) IM 1 :� 0, IM 2 :� 0, CM 1 :� 0, CM 2 :� 0 (2) for i � 1 to nr (3) for j � i to nc (4) s i :� sum of ith row of A, (5) s j :� sum of jth column of A.
end if (14) end for j (15)  Now, we provide graphs of Zagreb indices and coindices. Such type of graphical representation will be more helpful to study the dynamics of topological descriptors of the molecular graphs. Here, we present the strong double graph of the path graph, S D(P n ), where 2 ≤ n ≤ 21 and m � n − 1. In Figure 3, the behaviour of the first Zagreb index M 1 and second Zagreb index M 2 is linear as the straight plane, and the behaviour of the first Zagreb coindex CM 1 and second Zagreb coindex CM 2 is nonlinear as the curved form. In

Conclusion
We have presented generalized explicit formulas to calculate the first Zagreb index M 1 (G), second Zagreb index M 2 (G), first Zagreb coindex M 1 (G), and second Zagreb coindex M 2 (G) of the strong double graph S D(G) and k-iterated strong double graph SD k * (G). e relation between these indices and coindices is also presented as M 1 ≤ M 2 ≤ CM 1 ≤ CM 2 . We have also presented an algorithm with a given adjacency matrix to verify these indices and coindices by programming and numerically. Computergenerated graphs are also given to understand the dynamics of these indices and coindices.
is family of graphs can be considered for other degreebased and distance-related topological indices for further studies.

Data Availability
No data were used to support this study.

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

Authors' Contributions
All authors contributed equally to this work.