Abstract
Topological indices are graph-theoretic parameters which are widely used in the subject of chemistry and computer science to predict the various chemical and structural properties of the graphs respectively. Let be a graph; then, by performing subdivision-related operations , , , and on , the four new graphs (subdivision graph), (edge-semitotal), (vertex-semitotal), and (total graph) are obtained, respectively. Furthermore, for two simple connected graphs and , we define -sum graphs (denoted by ) which are obtained by Cartesian product of and , where . In this study, we determine first general Zagreb co-index of graphs under operations in the form of Zagreb indices and co-indices of their basic graphs.
1. Introduction
Graph theory has given different valuable tools in which likely the best tool is known as topological index (TI) that is used to predict structural and chemical properties of graphs such as connectivity, solubility, freezing point, boiling point, critical temperature, and molecular mass, see [1]. The medical behaviors and drugs’ particles of the different compounds are discussed with the help of various TIs in the pharmaceutical industries, see[2]. In addition, for the study of molecules, the quantitative structures’ activity relationships (QSAR) and quantitative structures’ property relationships (QSPR) are very useful techniques which are mostly performed with the help of TIs [3].
There are three basic types of TIs depending on the parameters of degree, distance, and polynomial. According to recent review [4], the degree-based TIs are mostly studied. First of all, Wiener calculated the boiling point of paraffin with the help of a degree-based TI, see [5]. Gutman and Trinajsti introduced Zagreb indices and used them to compute the different structure-based characteristics of the molecular graphs [6].
Later on, Shenggui and Huiling characterized the graphs for the first general Zagreb index [7]. Bedratyuk and Savenko calculated the ordinary generating function and linear recurrence relation for the sequence of the general first Zagreb index [8].
Recently, Ashrafi et al. defined Zagreb co-index and computed it for graphs which are formed using various operations, see [9, 10]. Kinkar et al. computed the first Zagreb co-indices of trees under different conditions, see [11]. Mansour and Song established relationship between Zagreb indices and co-indices of graphs [12]. Huaa and Zhang computed sharp bounds on the first Zagreb co-index in terms of Wiener, eccentric distance sum, eccentric connectivity, and degree distance indices [13]. Gutman et al. calculated relations between the Zagreb indices and co-indices of a graph and of its complement [14, 15].
In graph theory, the operations (union, intersection, complement, product, and subdivision) play an important role to develop new structure of graphs. Yan et al. computed the Wiener index for new graphs using five different operations , , , , and on a graph such as line graph , subdivided graph , line superposition graph , triangle parallel graph , and total graph , respectively, see [16]. After that, Eliasi and Taeri computed Wiener indices of newly defined -sum graphs represented as , where [17]. Furthermore, Deng et al. calculated Zagreb indices [18], Ibraheem et al. [19] forgot co-index, Liu et al. obtained first general Zagreb (FGZ) index [20], and Javaid et al. [21] computed the bounds of first Zagreb co-index; furthermore, they also studied the connection-based Zagreb index and co-index [22] of these graphs.
In this study, we computed FGZ co-index of graphs under operations such as , , , and . The reset of the study is settled as follows. Section 2 contains preliminaries. In Section 3, the main results of our work are discussed, and Section 4 has the conclusion of work.
2. Preliminaries
Let G be a simple and connected graph with vertex and edge set denoted by and , respectively. The degree of vertex any vertex in is the number of edges incident on it and denoted by d(v). Let G be a graph; then, its complement is defined as and iff denoted as . Gutman and Trinajsti introduced the first and second Zagreb indices as [6]
Ashrafi ȇt al. defined first Zagreb co-index as follows, see [10]:
Let be a graph; then, is obtained by adding one vertex in every edge of G.(i) is obtained from by inserting an edge between the vertices that are adjacent in G(ii) is obtained from by inserting an edge between new vertices that adjacent edges of G(iii)Apply both and on ; then, is obtained
Suppose two connected graphs and ; then, their graph is represented by having vertex set and iff and and , where .
For details, see Figure 1 and 2.
3. Main Results
This section contains results about FGZ co-index of graphs under operations.
Theorem 1. Let be -sum graph; then, its first general Zagreb co-index is given aswhere .
Proof. Using equation (2), we haveWe arrived at desired result by putting the values in equation (4).
Theorem 2. Let be -sum graph; then, its first general Zagreb co-index is given aswhere .
Proof. Using equation (2), we haveUsing equation 4, we directly haveWe arrived at desired result by putting the values in equation (6).
Theorem 3. Let be -sum graph; then, its first general Zagreb co-index is given aswhere .
Proof. Using equation (2), we haveUsing equation 4, we directly haveWe arrived at desired result by putting the values in equation (9).
Theorem 4. Let be -sum graph; then, its first general Zagreb co-index is given aswhere .
Proof. Using equation (2), we haveUsing equation 9, we directly haveThe value and are by equations (7) and (9)asfollows:By substituting the values of , , and in equation (12), we obtained required proof.
4. Conclusion
The study of the basic or factor graphs is an interesting problem in the theory of graphs where the original graphs becomes complex. In this study, we have computed FGZ co-index of graphs under operations such as , , , and in the terms of indices and co-indices of their basic or factor graphs.
Data Availability
The data used to support the findings of the study are included within article. However, for more details of the data can be obtained from the corresponding author.
Conflicts of Interest
The authors declare no conflicts of interest.
Authors’ Contributions
Muhammad Javaid and Uzma Ahmad contributed to the discussion of the problem, validation of results, final reading, and supervision; Muhammad Ibraheem contributed to the source of the problem, the collection of material, analyzing and computing the results, and initial drafting of the study; Ebenezer Bonyah and Shaohui Wang contributed to the discussion of the problem, the methodology, and the proofreading of the final draft.