Computing FGZ Index of Sum Graphs under Strong Product

Topological index (TI) is a function that assigns a numeric value to a (molecular) graph that predicts its various physical and structural properties. In this paper, we study the sum graphs (S-sum, R-sum, Q-sum and T-sum) using the subdivision related operations and strong product of graphs which create hexagonal chains isomorphic to many chemical compounds. Mainly, the exact values of first general Zagreb index (FGZI) for four sum graphs are obtained. At the end, FGZI of the two particular families of sum graphs are also computed as applications of the main results. Moreover, the dominating role of the FGZI among these sum graphs is also shown using the numerical values and their graphical presentations.


Introduction
Let G � (V(G), E(G)) be a (molecular) graph with V(G) and E(G) as sets of vertex and edge respectively. e degree of a vertex vεV(G) is the number of edges which are incident to v. A topological index (TI) is a function that assigns a numeric value to the under study (molecular) graph, see [1][2][3]. TIs are used to predict the various chemical and structural properties such as octane isomers including entropy, acentric factor, density, total surface area, molar volume, boiling point, capacity of heat at temperature and pressure, enthalpy of formation, connectivity of compounds and octanol water partition, see [4]. ese are also used to study the quantitative structure property and activity relationships which are very important in the subject of cheminformatics, see [5].
Wiener see [6] defined a distance-based TI to compute the boiling point of paraffin. Gutman and Trinajstic see [7] investigated total π-electron energy of the molecule graphs using a degree-based TI called by first Zagreb index nowadays. Later on, various Zagreb indices (hyper, forgotten, multiplicative and augmented) are defined to predict the different physicochemical properties chirality, heterosystems, complexity, branching and ZE-isomerism of the (molecular) graphs see [8]. Li and Zheng defined the first general Zagreb index (FGZI) and studied its different properties see [9]. e operations of graphs (addition, deletion, complement, product, union and intersection) also play a very important role for the construction of new graphs, see [10]. In particular, for a graph G Yan et al. [11] introduced the five new graphs S(G), R(G), L(G), Q(G) and T(G) are defined with the help of the operations of subdivision S, semitotal point R, Line graph L, semitotal edge Q and total vertex and edge T respectively. Also, Wiener index of these ϕ(G) graphs is computed, where Φ ∈ S, R, L, Q, T { }. Eliasi and Taeri [12] constructed the Φ-sum graphs G Φ + H by the Cartesian product of the graphs Φ(G) and H, where Φ(G) is obtained after applying the Φ on G for Φ ∈ S, R, Q, T { }. Moreover, they computed the Wiener indices of the Φ-sums graphs G S + H, G R + H, G Q + H and G T + H. Deng et al. [13] computed the first and second Zagreb indices of four operations on graphs using the concept of Cartesian product. Akhter and Imran [14] computed the forgotten topological index of four operations on graphs under Cartesian product. Liu et al. [15] computed the FGZI of Φ-sum graphs under the operation of Cartesian product. Sarala et al. [16] computed the F-index of Φ-sum graphs under the operation of strong product. For further study, we refer to [17][18][19][20][21][22][23][24][25][26][27][28][29].
In this paper, we extend this study and compute FGZI of the Φ-sums graphs (G Φ ⊠ H) under the operation of strong product of Φ(G) and H in term of FGZI of its factor graphs G and H, where Φ ∈ S, R, Q, T { }, G and H are any two connected graphs. e obtained results are general extension of the results of Deng et al. [13], Akhter and Imran [14], Liu et al. [15] and Sarala et al. [16] works. Forthcoming sections are arranged as: Section 2 includes the basic formulae and results, Section 3 covers the main results and Section 4 includes the conclusion.

Preliminaries
An atom is presented by a vertex and bonding between atom is showed by edge in the molecular graphs. e first and second Zagreb indices M 1 (G) and M 2 (G) of G are defined as follows [5,7,22]: For any real number α FGZ index and general Randic � index are defined as For more details of aforesaid TIs, see [30,31]. e following graphs are defined with the help of the subdivision related operations (S, R, Q and T), see [16]. In 1960 Sabidussi introduced the strong product for any two graphs G ∘ H has vertex set Cartesian product V(G) × V(H) such that (s 1 , t 1 ) and (s 2 , t 2 ) will be adjacent in G ∘ H iff: [s 1 � s 2 and t 1 is adjacent to t 2 ] or [t 1 � t 2 and s 1 is adjacent to s 2 ] or [s 1 is adjacent to s 2 and t 1 is adjacent to t 2 ]. Now, the Φ-sum graph under the operation of strong product is defined in [16].
Let G and H be two graphs. For Φ ∈ {S, R, Q, T}, Φ(G) is a graph constructed by the operation Φ on G with set of vertex V(Φ(G)) and set of edge E(Φ(G)).
en, Φ-sum graph G Φ ⊠ H under the strong product of graphs Φ(G) and H is a graph with vertex set It is observed that G Φ ⊠ H has |V(H)| copies of Φ(G) which are labeled by the vertices of H. Also, V(G) and E(G) are shown as blue and red vertices in G Φ ⊠ H respectively, see

Main Results
Now, we compute the analytical closed expression for FGZ index of the G S ⊠ H, G R ⊠ H, G Q ⊠ H and G T ⊠ H.

Theorem 1. Let G and H be two graphs. For α ∈ N + and
Proof. By the definition for any vertex For each vertex s 1 � s 2 ∈ V(G) and edge ( Proof. By definition Since d R(G) (s) � 2d G (s) For each vertex t 1 � t 2 ∈ E(H) and edge s 1 s 2 ∈ E(R(G)) where s 1 , s 2 ∈ (V(G)) then, Journal of Mathematics 5 d n s 1 , t 1 + d n s 2 , t 2 For each Proof. By definition For each vertex t 1 � t 2 ∈ V(H) and edge s 1 consider In C 1 , s 1 ∈ V(G) and d n (s 1 ) occurs d(s 1 ) times. us Let In C 1 , s 2 � uv ∈ E(G) and d n (s 2 ) occurs two times. erefore For each t 1 t 2 ∈ E(H) and (s 1 s 2 ) ∈ E(Q(G)), where s 1 , s 2 ∈ V(G), we have