Abstract
In theoretical chemistry, topological indices (TIs) have important role to predict various physical and structural properties of the study under molecular graphs. Among all topological indices, Zagreb-type indices have been used more effectively in the chemical literature. In this paper, we have computed first Zagreb, second Zagreb, forgotten, and hyper Zagreb indices of the generalized -sum graph in the form of different TIs of its basic graphs, where is a positive integer. This family of graphs is obtained by the lexicographic product of the graph and , where is constructed with the help of the generalized line superposition operation on . As a conclusion, we also checked the correlation between predefined graph under the operation of lexicographic product and with newly defined generalized -sum graphs using linear regression models of various degree-based TIs.
1. Introduction and Preliminaries
In the computing analysis of chemical compounds, the chemical structural formulas are usually represented by graphs. Mathematically, for a graph , the vertex and edge sets are denoted by and , respectively. The degree of the vertex (denoted by ) is the number of incident edges on it. The distance between any two vertices and in is denoted by and defined as the number of edges in a shortest path existing between the vertices and . Thoroughly, the graph is considered as a finite and simple graph.
The subject of chemical graph theory has many applications in chemistry. In a chemical graph, a vertex and an edge correspond to an atom and a chemical bond between them, respectively. A topological index is a function which presents a chemical graph in the form of a numerical number that is used to model the chemical and physical properties of molecules in quantitative structure property and activity relationships.
For a molecular graph , the first Zagreb index and second Zagreb index are firstly considered by Gutman and Trinajstic [1] in 1972 to study the total electron energy of the molecular graph which are defined as follows:
In 2015, forgotten index (F index) is defined as follows [2]:
In 2005, the first general Zagreb index is defined as follows [3]:
The concept of the general randic index (GRI) is defined by Bollobas and Erdos as
In 2013, hyper Zagreb index is defined by Shirdel et al. as follows [4]:
For more studies, we refer to [5]. In particular, we can find the results of TIs for various families of graphs such as nanosheets & nanostars dendrimers [6, 7], hex-derived networks [8], benzene networks [9, 10], and cellulose networks [11]. In addition, for the studies of the complex graphs, operations on graphs play a key role, where the original graphs are called the factors of the newly constructed graph under operations [12].
Yan et al. [13] defined the line superposition operation related to the subdivision of and computed the Wiener index of this derived graph , where is obtained by inserting one new vertex in every edge of and joining those pairs of new vertices by edges which have common adjacent (original) vertices (see Figure 1). Liu et al. extended this operation for any integral value of and obtained the generalized line superposition graph from the graph by inserting vertices in each edge and joining those pairs of new vertices by edges which have common adjacent (original) vertices [14] (see Figures 2 and 3).
Taeri et al. [15] constructed the -sum graph by Cartesian product of and , where is a line superposition graph, and and are assumed to be two connected graphs. Also, they computed the Wiener index of -sum graph. Whereas, Chu et al. [16], Deng et al. [17], Akhter et al. [18], and Liu et al. [14] calculated the different indices of -sum graph based on the Cartesian product. Recently, Liu et al. [19] extended this operation for any integral value of and obtained the generalized line superposition graph of the graph . Moreover, they constructed the generalized -sum graph and computed their Zagreb indices. Javaid et al. [20] constructed the generalized -sum graph based on strong product and computed its first and second Zagreb indices.
The composition or lexicographic product of two connected graphs and , which is denoted by , is a graph such that the set of vertices is , and two vertices and will be adjacent in if [ = , and is adjacent to ] or [ is adjacent to , and is adjacent to ].
Sarala et al. [21], De [22], Pattabiraman [23], Pattabiraman and Santhakumar [24], and Suresh and Devi [25] have computed different degree-related Zagreb indices for the graphs based online superposition operation and lexicographic product.
(1) Qα-sum: let and be two graphs; first, we apply generalized line superposition operation on to get graph. Then, we take lexicographic product between and to get -Sum graph denoted by . -sum graph having vertex-setsuch that two vertices and of are adjacent if [ = in , and is adjacent to in ] or [ is adjacent to in , and is adjacent to in ], where is a positive integer (see Figure 4).
To check the correlation between the predefined lexicographic product of two simple graphs and newly defined -sum graphs , one may use the linear regression when having a linear relationship between the dependent variable ( =) and the independent variable ( =). For further studies related to operations and graph products, the readers are referred to [26–29]. In this paper, we have used the generalized line superposition operation to construct a new generalized line superposition graph from the graph actual graph . Then, we have constructed a -sum graph based on the lexicographic product of and , denoted by . Moreover, we have computed the first Zagreb, second Zagreb, forgotten, and hyper Zagreb indices of in terms of its factor graphs and . We have extended this work by constructing linear regression models between and -sum graphs via foresaid degree-based TIs.
2. Main Results
This section covers the main results.
Theorem 1. Let and be two connected graphs such that , . For ,
Proof. Now,For which is inserted into the edge of , we have . This givesSo, we haveNow, we takewhere and are the vertices that are inserted into the edges of and of , respectively.Consequently, we have obtained our required result.
Theorem 2. Let and be two connected graphs such that , . For ,where is the number of neighbours which are common between and in .
Proof. Now, we takeSo,where is the vertex inserted into the edge of where is the set of neighbor vertices of in where is the added vertex in the edge , and is the added vertex in the edges of where is the number of neighbors which are common vertices of and in . Consequently, we have obtained our required result.
Theorem 3. Let and be two connected graphs such that , . For ,
Proof.
ConsiderNow,Now, we takeFor that is introduced into the edges of , we have . This givesSo,where and are the nodes that are introduced into the edge set of and of Consequently, we have obtained our required result.
Theorem 4. Let and be two connected graphs such that , . For ,
Proof. Now,Now, we takeOne can see that for a vertex , , where . Thus, and are the nodes of , so we haveConsequently, the result is over.
3. Applications and Conclusion
For any two simple path graphs and with , we construct their lexicographic product graphs and compute various degree-related TIs, as shown in Table 1 with certain values of and . Furthermore, we construct by applying generalized line superposition operation on and then construct -sum graph based on lexicographic product of and . Finally, we compute certain degree-related TIs using Theorem 1–Theorem 4, as given in Table 2. We note that by the addition of vertices , the values of obtained TIs are increasing for the newly constructed graphs comparatively to the graphs obtained by the ordinary lexicographic product of graphs.
3.1. Linear Regression Model
To check the correlation between the predefined lexicographic product of two simple graphs and newly defined -sum graphs , we have made the linear regression model for all the obtained indices as follows:
3.2. First Zagreb Index
where Y = index values, is the constant, is the regression coefficient, and X = index values (see Tables 1 and 2).
3.3. Second Zagreb Index
where Y = index values, is the constant, is the regression coefficient, and X = index values (see Tables 1 and 2).
3.4. Forgotten Topological Index
where Y = index values, is the constant, is the regression coefficient, and X = index values (see Tables 1 and 2).
3.5. Hyper Zagreb Index
where Y = index values, is the constant, is the regression coefficient, and X = index values (see Tables 1 and 2).
Now, we represent a tabular representation of the correlation and square of correlation coefficient values related to the TI’s of certain type of graphs and (see Tables 3 and 4). The highest value of and are indicated in bold.
For more explanation, see the graphical representation of correlation and square of correlation coefficient in Figures 5 and 6.
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.