Bounds of Degree-Based Molecular Descriptors for Generalized F -sum Graphs

A molecular descriptor is a mathematical measure that associates a molecular graph with some real numbers and predicts the various biological, chemical, and structural properties of the underlying molecular graph. Wiener (1947) and Trinjastic and Gutman (1972) used molecular descriptors to ﬁnd the boiling point of paraﬃn and total π -electron energy of the molecules, respectively. For molecular graphs, the general sum-connectivity and general Randi´c are well-studied fundamental topological indices (TIs) which are considered as degree-based molecular descriptors. In this paper, we obtain the bounds of the aforesaid TIs for the generalized F -sum graphs. The foresaid TIs are also obtained for some particular classes of the generalized F -sum graphs as the consequences of the obtained results. At the end, 3 D -graphical presentations are also included to illustrate the results for better understanding.


Introduction
A molecular descriptor called by the topological index (TI) is a function from the set of (molecular) graphs to the set of real numbers. TIs are studied as a subtopic of chemical graph theory to predict the chemical reactions, biological attributes, and physical features of the compounds in theoretical chemistry, toxicology, pharmaceutical industry, and environmental chemistry, see [1]. In addition, these TIs are also used to characterize the molecular structure with respect to quantitative structure activity and property relationships which are studied in the subject of cheminformatics, see [2]. TIs have been classified into different classes but degree-based TIs play a significant part in the theory of chemical structures or networks. Firstly, Wiener [3] used the TI to find boiling point of paraffin. Gutman and Trinajstic calculated total π-electron energy of the molecules by a TI that is recognized as the first Zagreb index in the literature, see [4]. In 2009, Zhou and Trinajstic [5] suggested the sum-connectivity index, which was subsequently generalized in 2010 [6]. e Randić index was defined in 1975 [7]; later on the idea was subsequently extended to the generalized Randić connectivity index by Li and Gutman, see [8]. For more studies, we refer to [9][10][11][12].
In chemical graph theory, operations of graphs are frequently used to find the new families of graphs. Yan et al. [13] defined the four subdivision-related operations (S 1 1 , S 2 1 , S 3 1 , and S 4 1 ) on a molecular graph M and obtained the Wiener indices of the resultant graphs S 1 1 (M), S 2 1 (M), S 3 1 (M), and S 4 1 (M). After that, Eliasi and Taeri [14] defined the F-sum graph M 1 + F 1 M 2 with the help of Cartesian product of M 1 and F(M 2 ), where F 1 ϵ S 1 1 , S 2 1 , S 3 1 , S 4 1 . Deng et al. [15] and Akhter and Imran [16] also computed the 1st and 2nd Zagreb and general sum-connectivity indices of the F-sum graphs, respectively. Recently, Liu et al. [17] generalized these subdivision-related operations and defined the generalized F-sum graphs where k ≥ 1 is an integral value. ey also computed the 1st and 2nd Zagreb indices for these newly obtained graphs. For further studies of F-sum and generalized F-sum graphs, see [18][19][20][21][22][23][24][25][26][27]. Now, we extend this study by computing the bounds (upper and lower) of the general sum-connectivity and general Randić indices for the generalized F-sum graphs. In the remaining paper, Section 2 consists of main results of bounds and Section 3 has conclusion and applications of the obtained results.
Liu [17] defined the following graphs using the generalized subdivision-related operations: Definition 2. Let M 1 and M 2 be two connected graphs, F k ε S 1 k , S 2 k , S 3 k , S 4 k and F k (M 1 ) be a graph (obtained after applying the operation F k on M 1 with vertex set V(F k (M 1 )) and edge set E(F k (M 1 )). en, the generalized F-sum graph For more details, see Figures 2 and 3.

Main Results
In this section, we find out the sharp bounds of GSCI and GRI of generalized F-sum graphs.
where equalities hold if M 1 and M 2 are regular graphs.
Proof. (a) By the definition of GSCI, we have Discrete Dynamics in Nature and Society (2) (1, a) (4, c) Since in this case |E(S 1 Consequently, where equalities hold if M 1 and M 2 are regular graphs. □ Proof. (b) By the definition of GRI, we have Discrete Dynamics in Nature and Society 5 Since Since in this case |E(S 1 Consequently,  Proof. (a) By the definition of the GSCI, we have Consider for Discrete Dynamics in Nature and Society Hence, Equality holds if M 1 and M 2 are regular graphs.

□
Proof. (b) By the definition of GRI, we have 8 Discrete Dynamics in Nature and Society Consider for Discrete Dynamics in Nature and Society Hence, Equality holds if M 1 and M 2 are regular graphs.
10 Discrete Dynamics in Nature and Society Now, e graphical representation of Example 4(a) is depicted in Figure 10. e lower bounds are represented by the blue graph and the upper bounds are represented by the green graph.
e graphical representation of Example 4(b) is depicted in Figure 11. e lower bounds are represented by the yellow graph and the upper bounds are represented by the blue graph.

Conclusion
In this paper, the lower and upper bounds of the general sumconnectivity and general Randić indices of the generalized Fsum graphs (F k -sum graphs) are computed in terms of the order, size, maximum, and/or minimum degree and Zagreb indices of the factor graphs, where the F k -sum graphs are obtained with the help of four generalized subdivision-related operations and the Cartesian product of graphs. However, the problem is still open for other types of product of graphs.    Figure 11: Graph for α � 2 2 4 2 (nm − n) + 2 2 2 .3 2 (mn − m) + 24 2 (mn − m) + 6m (4m − 6) 2 + 24 2 m ≤ R α (P n + S 4 4 P m ) ≤ 2 2 2 (nm− n)+ 6 2 (mn − m) + 22 2 (mn − m) + 6m(4m − 6) 2 + 4 2 m. 16 Discrete Dynamics in Nature and Society

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.