Abstract
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 -electron energy of the molecules, respectively. For molecular graphs, the general sum-connectivity and general Randić 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 -sum graphs. The foresaid TIs are also obtained for some particular classes of the generalized -sum graphs as the consequences of the obtained results. At the end, -graphical presentations are also included to illustrate the results for better understanding.
1. 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 -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]. The 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 [9–12].
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 on a molecular graph and obtained the Wiener indices of the resultant graphs , and . After that, Eliasi and Taeri [14] defined the -sum graph with the help of Cartesian product of and , where . Deng et al. [15] and Akhter and Imran [16] also computed the 1st and 2nd Zagreb and general sum-connectivity indices of the -sum graphs, respectively. Recently, Liu et al. [17] generalized these subdivision-related operations and defined the generalized -sum graphs for , where is an integral value. They also computed the 1st and 2nd Zagreb indices for these newly obtained graphs. For further studies of -sum and generalized -sum graphs, see [18–27].
Now, we extend this study by computing the bounds (upper and lower) of the general sum-connectivity and general Randić indices for the generalized -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.
2. Preliminaries
Throughout, we consider undirected, connected, and simple graphs with and as vertex and edge sets, respectively. In addition, and are the order and size of M. For a vertex and as a degree of in , and are the maximum and minimum degrees of the graph . The graphs and and are called path , cycle , and complete , respectively. For further basic terminologies, see [28].
Definition 1. For a real number and (molecular) graph , the general sum-connectivity index (GSCI) and general Randić index (GRI) areMoreover, and then are known as the 1st Zagreb index and hyper-Zagreb index. If , then ; it is known as Randić, 2nd Zagreb, and reciprocal Randić indices.
Liu [17] defined the following graphs using the generalized subdivision-related operations:(i) graph is obtained by inserting -vertices in each edge of .(ii) is obtained from by joining the old vertices which are adjacent in .(iii) is obtained from by joining the new vertices lying in an edge to the corresponding new vertices of the other edge, if these edges have some common vertexes in .(iv) is the union of and graphs.For more details, see Figure 1 for .
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Definition 2. Let and be two connected graphs, and be a graph (obtained after applying the operation on with vertex set and edge set . Then, the generalized -sum graph is a graph with vertex set in such a way that are adjacent if [ and ] or [ and ], where is an integral value.
For more details, see Figures 2 and 3.
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3. Main Results
In this section, we find out the sharp bounds of GSCI and GRI of generalized F-sum graphs.
Theorem 1. For a real number and counting number , the lower and upper bounds on the GSCI and GRI of generalized F-sum graph are as follows:(a)(b)where equalities hold if and are regular graphs.
Proof. (a) By the definition of GSCI, we haveConsiderSince and , we haveSince in this case , we haveConsequently,where equalities hold if and are regular graphs.
Proof. (b) By the definition of GRI, we haveConsiderSince and , we haveSince in this case , we have .
Consequently,where equalities hold if and are regular graphs.
Theorem 2. For a real number and counting number , the lower and upper bounds on the GSCI and GRI of the generalized F-sum graph are as follows:(a)(b)where equalities hold if and only if and are regular graphs.
Proof. (a) By the definition of the GSCI, we haveConsiderConsider for , we have if ; for , we obtain and for , we have .
Now, considerNote that and if , then , and if , then .Hence,Equality holds if and are regular graphs.
Proof. (b) By the definition of GRI, we haveConsiderConsider for , we have if ; for , we obtain and for , we have . Now, considerNote that and if , then , and if , then .Hence,Equality holds if and are regular graphs.
Theorem 3. For a real number and counting number , the lower and upper bounds on the GSCI and GRI of the generalized F-sum graph are as follows:(a)(b)where equalities hold if and only if and are regular graphs.
Proof. (a) By the definition of the GSCI, we haveNow,Now,Note that for , where is the vertex inserted into the edge of . Then, we haveWe break this sum into two parts , where belong the edges and belong the .Since is the vertex inserted into the edge of and is the vertex inserted into the edge of ,Consequently, we have
Proof. (b) By the definition of the GRI, we haveNow,Now,Note that for , where is the vertex inserted into the edge of . Then, we haveWe break this sum into two parts , where belong the edges and belong the .Since is the vertex inserted into the edge of and is the vertex inserted into the edge of ,Consequently, we have
Theorem 4. For a real number and counting number , the lower and upper bounds on the GSCI and GRI of the generalized -sum graph are(a)(b)where is the number of common neighbors of and in .
Proof. The proof follows by Theorem 2 and Theorem 3.
4. Applications
For a real number , the lower and upper bounds on GSCI and GRI of the generalized -sum graphs obtained by the particular classes of graphs as the consequences of the obtained results are given in Figures 4–11.
Example 1. For , , and , we have(a), and(b)The graphical representation of Example 1(a) is depicted in Figure 4, the lower bounds are represented by the blue graph and the upper bounds are represented by the red graph. The graphical representation of Example 1(b) is depicted in Figure 5, the lower bounds are represented by the green graph and the upper bounds are represented by the blue graph.
Example 2. For , , and , we have(a)(b)The graphical representation of Example 2(a) is depicted in Figure 6. The lower bounds are represented by the Niagara Azure graph and the upper bounds are represented by the gold graph. The graphical representation of Example 2(b)is depicted in Figure 7. The lower bounds are represented by the pink graph and the upper bounds are represented by the yellow graph.
Example 3. For , , and , we have(a)(b)The graphical representation of Example 3(a) is depicted in Figure 8. The lower bounds are represented by the blue graph and the upper bounds are represented by the green graph. The graphical representation of Example 3(b) is depicted in Figure 9. The lower bounds are represented by the yellow graph and the upper bounds are represented by the gray graph.
Example 4. For , , and , we have(a)(b)The graphical representation of Example 4(a) is depicted in Figure 10. The lower bounds are represented by the blue graph and the upper bounds are represented by the green graph. The graphical representation of Example 4(b) is depicted in Figure 11. The lower bounds are represented by the yellow graph and the upper bounds are represented by the blue graph.
5. Conclusion
In this paper, the lower and upper bounds of the general sum-connectivity and general Randić indices of the generalized F-sum graphs (-sum graphs) are computed in terms of the order, size, maximum, and/or minimum degree and Zagreb indices of the factor graphs, where the -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.
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
The authors have no conflicts of interest.
Acknowledgments
This research was funded by the Project of Anhui Jianzhu University under Grant nos. 2016QD116 and 2017dc03.