Abstract
There are various methods available which are used to search large chemical databases and to predict the physicochemical properties of molecular structures. Using molecular descriptors for this purpose is the simplest of these methods. The Zagreb indices are amongst the oldest molecular descriptors, and their properties have been extensively studied and applied in QSAR/QSPR studies. The Zagreb coindices were recently introduced, attracting the attention of researchers in mathematical chemistry. In this paper, we study Zagreb indices and several other Zagreb-type indices including the general Randić index, sum-connectivity index, F-index, and Zagreb coindices of -vertex and edge join of two arbitrary graphs.
1. Introduction
The relation between the atoms and chemical bonds between atoms in a molecule can be represented by a molecular graph, commonly denoted by . The atoms and their chemical bonds, respectively, play the role of vertices and edges of the corresponding molecular graph [1]. A graph is a pair with vertex set and edge set . The order and size of are, respectively, the cardinalities and . The neighbourhood of a vertex in is the set of vertices adjacent to . The degree of a vertex in is defined as . Let and , respectively, denote the path and cycle graphs of order . For simplicity, we use the notation instead of . Other important and undefined terminologies from graph theory can be found in [2].
A scalar quantity, which is necessarily a graph invariant, which can be used to represent a certain topological feature of a molecular graph is known as a topological index [3, 4]. These indices are used as functions assigning numerical values to the qualitative properties of molecules. The structural properties of molecules have impact on their physicochemical properties. These properties are correlated by using topological indices in QSAR/QSPR studies [5]. In QSAR/QSPR studies, these indices are used in search for new chemicals which are then used in producing potent drugs [6–9]. These topological indices are also used for virtual screening of chemical libraries to search for new chemicals of specific needs in computational chemistry and pharmacology [10–13]. The first topological index was used as a numerical descriptor to correlate the boiling points of some paraffins by Wiener [14] in 1947. This descriptor was called Wiener number, but Hosoya [15] used the term topological index for the first time in 1971. He defined the Wiener index of a graph as follows:
Gutman and Trinajstć [16, 17] investigated the dependence of the total -electron energy of an alternant hydrocarbon, where they encountered the terms denoted by and commonly known as Zagreb indices of the first and second kind and defined as follows:
Randić [18] introduced the Randić connectivity index in 1975 as follows:
Bollobás and Erdos [19] defined the general Randić index as follows:
The general sum-connectivity index was defined by Zhou and Trinajstić [20] as follows:
Furtula and Gutman [21] introduced the forgotten topological index as
Došlić [22] defined the first and second Zagreb coindices as follows:
Some details and applications of these indices can be found in [23–27] and a survey paper on the Randić index in [28]. The zeroth-order general Randić index is a general case of the first Zagreb index [29, 30]. Zagreb indices of some tree-like graphs containing hydrocarbons like ethane, propane, and butane are studied in [31].
In [32], the authors studied the first and second Zagreb indices of some graph operations including subdivision and total graphs. The Zagreb coindices were recently studied in [33–37], and details on the relations between Zagreb indices and coindices can be found in [38–44]. In [45], the upper and lower bounds of and are determined. An upper bound of in -apex trees is also studied, and the corresponding extremal -apex trees are also characterized. Two distance-based indices of some graph operations were studied in [46].
2. -Graphs
The -graph [47] of a graph (denoted by ) is the graph obtained from by adding a vertex corresponding to every edge and joining to end vertices and for each edge . Let denote the set of all new vertices , that is,
Similarly, let denote the new edges in , that is,
Then, we have
The degree of vertex is given by
Based on graph , two new graph operations obtained from two arbitrary graphs and were defined in [48]. These operations are called -vertex join and -edge join of and , respectively, denoted by and . For two vertex disjoint graphs and , the -vertex join of and is obtained from and by joining every vertex of and by an edge. Similarly, for two vertex disjoint graphs and , the -edge join of and is obtained from and by joining every vertex of and by an edge. Graphs and are shown in Figure 1.
3. -Vertex Join and -Edge Join of Graphs
Let and be two graphs. The order and size of are denoted by and , . Let and , respectively, denote graphs and . The vertex and edge sets of graphs and are given as follows:where . Similarly,where .
Then, the order and size of graph can be observed as and . Similarly, the order and size of graph is given by and . The order and size of the complements of the sets and are given by
The degree of a vertex in and can be obtained from the following expressions:
The following lemma will be used for calculating the Zagreb coindices of raphs and .
Lemma 1 (see [33]). The first and second Zagreb coindices of graph of order and size can be computed as
In the next section, we compute the Zagreb indices of the graph and -vertex and -edge join of graphs and the Zagreb coindices of and -vertex and -edge join of graphs.
4. Zagreb Indices of -Graphs and -Vertex and Edge Join of Graphs
Theorem 1. The first Zagreb index of graph is given by
Proof. Using the definition of the vertex set of graph and by using (2), we get
Theorem 2. The second Zagreb index of graph is given by`
Proof. Using the definition of the edge set of graph and by using (3), we get
Theorem 3. The first Zagreb index of graph is given by
Proof. The definition of the vertex set of graph and the use of (2) and (3) yield the following:
Theorem 4. The second Zagreb index of graph is given as
Proof. Using the definition of the edge set of graph and by using (2) and (3), we get the following:
Theorem 5. The first Zagreb index of graph is calculated as
Proof. Using the vertex set of graph and by using (2) and (3), we get the following:
Theorem 6. The second Zagreb index of graph is given as
Proof. Using the edge set of graph and by using (2) and (3), we get
It can be observed that, for graph and , we have . Using this relation and the definitions of Zagreb coindices, we obtain the following results.
Theorem 7. The first Zagreb coindex of graph is given by
Proof. Using Lemma 1, Theorem 3, and (8), we obtain the following:
Theorem 8. The first Zagreb coindex of graph is given as
Proof. By substituting Theorem 5 in (8) and using Lemma 1, we obtain
Theorem 9. The second Zagreb coindex of graph is as follows:
Proof. Using (9) and substituting Theorems 3 and 4 in Lemma 1, we obtain
Theorem 10. The second Zagreb coindex of graph is given as
Proof. Using (9) and substituting Theorems 5 and 6 in Lemma 1, we obtain
5. General Randić Index of -Vertex and Edge Join of Graphs
In this section, we compute the general Randić index , for , of the -vertex join and -edge join of graphs.
Theorem 11. The general Randić index , for , of is given as
Proof. Using (4) and the edge set of , we obtain the following:
Theorem 12. General Randić index of graph for is
Proof. Using (4) and the edge set of , we obtain the following:
6. Forgotten Index of -Vertex and Edge Join of Graphs
In this section, we calculate the forgotten index of the -vertex and -edge join of graphs.
Theorem 13. The forgotten index of is given as
Proof. Using (7) and the edge set of , we get
Theorem 14. The forgotten index of graph is given as follows:
Proof. Using (7) and the edge set of , we get
7. General Sum-Connectivity Index of -Vertex and Edge Join of Graphs
In this section, we obtain the general sum-connectivity index of the -vertex and -edge join of graphs.
Theorem 15. The general sum-connectivity index of graph is given by
Proof. Using (6), we getWhen , substituting expressions from Theorems 13 and 4, we getSimilarly, when , we obtain by substituting expressions from Theorems 14 and 6 as follows:
Data Availability
The data used to support the findings of this study are included within the article.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.
Acknowledgments
The research work of the first author was sponsored by the Shandong Provincial Natural Science Foundation, China (ZR2018MA010). This research was also supported by UPAR Grants of United Arab Emirates University (UAEU), UAE, via Grant nos. G00002590 and G00003271.