First General Zagreb Index of Generalized F-sum Graphs

)e first general Zagreb (FGZ) index (also known as the general zeroth-order Randić index) of a graph G can be defined as M(G) � 􏽐uv∈E(G)[d c− 1 G (u) + d c− 1 G (v)], where c is a real number. As M(G) is equal to the order and size of G when c � 0 and c � 1, respectively, c is usually assumed to be different from 0 to 1. In this paper, for every integer c≥ 2, the FGZ index M is computed for the generalized F-sums graphs which are obtained by applying the different operations of subdivision and Cartesian product. )e obtained results can be considered as the generalizations of the results appeared in (IEEE Access; 7 (2019) 47494–47502) and (IEEE Access 7 (2019) 105479–105488).


Introduction
Graph theory concepts are being utilized to model and study the several problems in different fields of science, including chemistry and computer science. A topological index (TI) of a (molecular) graph is a numeric quantity that remained unchanged under graph isomorphism [1,2]. Many topological indices have found applications in chemistry, especially in the quantitative structure-activity/property relationships studies; for detail, see [3][4][5][6][7][8][9][10][11][12][13].
Wiener index is the first TI introduced by Harry Wiener in 1947, when he was working on the boiling point of paraffin [14]. In 1972, Trinajstić and Gutman [15] obtained a formula concerning the total energy of π electrons of molecules where the sum of square of valences of the vertices of a molecular structure was appeared. is sum is nowadays known as the first Zagreb index. In this paper, we are concerned with a generalized version of the first Zagreb index, known as the general first Zagreb index as well as the general zeroth-order Randić index.
ere are several operations in graph theory such as product, complement, addition, switching, subdivision, and deletion. In many cases, graph operations may be helpful in finding graph quantities of more complicated graphs by considering the less complicated ones. In chemical graph theory, by using different graph operations, one can develop large molecular structures from the simple and basic structures. Recently, many classes of molecular structures are studied with the assistance of graph operations.
In 2007, Yan et al. [6] listed the five subdivision operations with the help of their vertices and edges. ey also discussed the different features of Wiener index of graphs under these operations. After that, Eliasi and Taeri [16] introduced the F 1 -sum graphs Γ 1+F 1 Γ 2 with the assistance of Cartesian product on graphs F 1 (Γ 1 ) and Γ 2 , where F 1 (Γ 1 ) is obtained by applying the subdivision operations S 1 , R 1 , Q 1 , and T 1 . ey also defined the Wiener indices of these resulting graphs Γ 1+S 1 Γ 2 , Γ 1+R 1 Γ 2 , Γ 1+Q 1 Γ 2 , and Γ 1+T 1 Γ 2 . Later on, Deng et al. [17] calculated the 1st and 2nd Zagreb topological indices, and Imran and Akhtar [18] calculated the forgotten topological index of the F 1 -sums graph. In 2019, Liu et al. [19] computed the first general Zagreb index of F 1 -sums graphs.
e remaining work is arranged as follows: Section 2 contains some basic definitions, Section 3 contains the key outcomes, and Section 4 contains the some particular applications. Conclusions of the obtained results are presented in Section 5.

Preliminaries
Let Γ � (V(Γ), E(Γ)) be a simple graph having |V(Γ)| the order and |E(Γ)| the size of a graph, where V(Γ) is considered as node set and E(Γ)⊆V(Γ) × V(Γ) is a bond set. Every vertex is considered as an atom in a graph, and bonding within the two atoms is known as edge. e valency or degree of any node is the number of total edges which are incident to the node. Now, few useful TI's are explained given below: Definition 1. If Γ be a connected graph, then the 1st and 2nd Zagreb topological indices as (1) ese two descriptors of the graph were introduced by Trinajsti and Gutman [15]. Such type of TI's have been utilized to discuss the QSAR/QSPR of the different chemical structures such as chirality, complexity, hetero-system, ZEisomers, π electron energy, and branching [9,10].

Definition 2.
If R is the real number, c ∈ R − 0, 1 { }, and Γ be a connected graph, so the 1st general Zagreb topological index is given as Definition 3. If R is the real number, c ∈ R, and Γ be a connected graph, so the general Randic � is given as where R − (1/2) is considered as the classical Randic � connectivity topological index. e generalized F-sums graph is defined in [20] as follows: vertices lying on edge to the corresponding new vertices of other edge, if these edges have some common vertex in G. (iv) T k (G) is union of R k (G) and Q k (G) graphs. For further details, see Figure 1.

Definition 4.
If Γ 1 &Γ 2 be two connected molecular structures, F k ∈ S k , R k , Q k , T k and F k (Γ 1 ) be a structure obtained after using F k on Γ 1 with bonds (edges) E(F k (Γ 1 )) and nodes (vertices) V(F k (Γ 1 )). So, the generalized F-sums graph (Γ 1+F k Γ 2 ) is a structure with nodes: in such a way two nodes ( Discrete Dynamics in Nature and Society
Proof. en by definition, we have, For α � c − 1, the above equation is consider as Discrete Dynamics in Nature and Society 6 Discrete Dynamics in Nature and Society For every vertex b ∈ V(Γ 2 ) & edge ac ∈ E(R k (Γ 1 )) a, c ∈ V(Γ 1 ), then the 2nd term of (16) will be b∈V Γ 2 For . So the 3rd term of (16) will be Discrete Dynamics in Nature and Society and the 4th term of (5) is Since in this case |E(S k (Γ 1 ))| � (k − 1)|e Γ 1 , we have Using (17), (18), (20), and (22) in (16), then we have □ Theorem 3. Let Γ 1 and Γ 2 be two simple graphs and where N is the set of natural numbers and α � c − 1.
Proof. en by definition, we have 8 Discrete Dynamics in Nature and Society , then the 1st term of (25) will be (26) Discrete Dynamics in Nature and Society 9 Now ∀ b ∈ V(Γ 2 ), ac ∈ E(Q k (Γ 1 )) if a ∈ V(Γ 1 ) and c ∈ V(Q k (Γ 1 )) − V(Γ 1 ); the 1st term of (27) will be b∈V Γ 2 . en the 2nd term of equation (27) splits into two parts for the vertices a and c, then the equation will be 10 Discrete Dynamics in Nature and Society Using (26), (28), (29), and (30) in (25), we get the required result: Theorem 4. Let Γ 1 and Γ 2 be two simple graphs. e FGZ index of the generalized T-sum graph Γ 1 + T k Γ 2 is
Theorem 5. Assume that Γ 1 and Γ 2 are two simple graphs and α � c − 1, where c ∈ R − 0, N + { } and R is a set of real number. en, the FGZ index of generalized F-sum graphs   Table 2.
From Figure 6, it is clear that the behavior of FGZ index of the generalized Q-sum graph Γ 1 + Q k Γ 2 at t � 2 is more better than t � 0 and t � 1: (38) From Figure 7, it is clear that the behavior of FGZ index of the generalized T-sum graph Γ 1 + T k Γ 2 at t � 0 is more better than t � 1 and t � 2.

Conclusions
Now, we close our discussion with the following remarks: (i) For positive integer k and two graphs Γ 1 & Γ 2 , we have computed FGZ index of the generalized F-sums graphs Γ 1+F k Γ 2 , where generalized F-sums graphs are obtained by the different operations of subdivision and Cartesian product on Γ 1 & Γ 2 .
(ii) e obtained results are also verified and illustrated for the particular classes of graphs.
(iii) e behavior of FGZ index is also analyzed with the help of numerical and graphical presentations.
(iv) However, the problem is still open to compute the different topological indices (degree and distance based) for the generalized F-sum graphs.

Data Availability
All the data are included within this paper. However, the reader may contact the corresponding author for more details of the data.

Conflicts of Interest
e authors have no conflicts of interest. t layer t_0 t_2 t_-1 Figure 6: Numerical behavior of M c (P m+Q k P n ) using Table 3. Discrete Dynamics in Nature and Society