Abstract
A topological index (TI) is a molecular descriptor that is applied on a chemical structure to compute the associated numerical value which measures volume, density, boiling point, melting point, surface tension, or solubility of this structure. It is an efficient mathematical method in avoiding laboratory experiments and time-consuming. The forgotten coindex of a structure or (molecular) graph is defined as the sum of the degrees of all the possible pairs of nonadjacent vertices in . For and the connected graph , the derived graphs are obtained by applying the operations (subdivided), (triangle parallel), (line superposition), and (total graph), respectively. Moreover, a derived sum graph (-sum graph) is obtained by the Cartesian product of the graph with the graph . In this study, we compute forgotten coindex of the -sum graphs (-sum), (-sum), (-sum), and (-sum) in the form of various indices and coindices of the factor graphs and . At the end, we have analyzed our results using numerical tables and graphical behaviour for some particular -sum graphs.
1. Introduction
Chemical graph theory being the combination of graph theory and chemistry is a branch of mathematical chemistry in which we study the various physical and chemical properties of a chemical structure or network using different graph theoretical techniques. A topological index is one of the most used graph theoretical technique that studies the different properties of the molecular structure such as volume, density, freezing point, vaporization point, boiling point, melting point, surface tension or solubility, heat of formation, and heat of evaporation numerically [1, 2].
TIs are categorized in three types such as degree, distance, and polynomial based, but according the latest survey [3], the degree-based TIs are more studied than others. Wiener was the first scientist who calculated the boiling point of paraffin by using a distance-based TI [4]. Gutman and Trinajsti [5] introduced degree-based TIs known as Zagreb indices to calculate total -electron energy of hydrocarbons. Rouvary [6] and Balaban [7, 8] discussed the structure-activity correlations of different chemical phenomenon using TIs. Klein et al. introduced a molecular topological index and drive a close relation with the Wiener index [9, 10]. Mendiratta et al. and Cornwell used Wiener’s TI to study structure-activity study on antiviral 5-vinylpyrimidine nucleoside analogs [11]. Gutman and Estrada calculated the TIs based on the line graph of the molecular graph [12]. Biye et al. wrote a novel on TI for QSPR/QSAR study of organic compounds [13]. Qinghua et al. calculated the TI for octane number of acyclic alkane by autocorrelation [14]. Baig et al. computed the TIs for polyoxide and silicate DSL and DOX-like networks [15].
In 2015, Furtula and Gutman redefined a TI with the name forgotten index (-index) as the sum of cubes of vertex degrees of a molecular graph [16]. Gutman et al. listed the graphs having smallest forgotten index [17]. Zhongyuan and Zhibo computed bounds of the -index using Cauchy-Schwarz inequality, Jensen’s inequality, and Chebyshev’s sum inequality [18]. Gao and Liu calculated -index of different chemical structures [19, 20]. Ahmad et al. worked online graph of benzene ring in the 2D network and calculated the degree-based TIs for these graphs [21]. Javaid et al. calculated bound on -index for unicyclic graphs with fixed number of pendent vertices [22]. Imran et al. [23] investigated the family of unicyclic graphs with extreme -coindex.
Ashrafi et al. introduced the concept of coindex of graph and investigated Zagreb coindices of composite graph operations such as union, disjunction, and various product of graphs [24]. Havare et al. computed the -index and -coindex for carbon base nanomaterial [25]. Basavanagoud and Desai calculated the -index and hyper-Zagreb index of generalized transformation graphs [26]. Gao et al. calculated electron energy of molecular structures -index [27]. Yasir et al. computed -index of dendrimers-like structure [28]. Basavanagoud and Timmanaikar calculated first Zagreb and -index of some dominating transformation graphs [29]. Sana et al. characterized the extremal graphs and proved the ordering among the different subfamilies of graphs with respect to -index [30].
Various operations on a graph play the important role in the development of different new classes of graphs. Yan et al. listed five graphs line graph , subdivided graph , line superposition graph , triangle parallel , and total by performing different operations on and computed Wiener indices of these five graphs [31]. Eliasi and Taeri defined the derived sum graphs and computed their Wiener indices, where [32]. Later on, for these derived sum graphs, Deng et al. [33] computed first two Zagreb indices, and Akhtar and Imran calculated the -index [34, 35]. Javaid et al. (2021) investigated the first Zagreb connection index and coindex [36], and Javaid et al. investigated forgotten index of these graphs [37]. Moreover, Pattabiraman and Peng computed -indices and their coindices of some classes of graphs [38, 39]. Ali et al. forgotten coindex of some nontoxic dendrimers structure used in targeted drug delivery [40].
In this study, we compute forgotten coindices of -sum graphs , where in the form of forgotten index, first Zagreb indices, and coincides of their basic graphs and . At the end, the obtained results are also illustrated with the assistance of the examples for some particular -sum graphs. In Section 2, the definitions and notations are presented, Section 3 includes the main results of our work, and Section 4 presents particular examples related to the main results.
2. Preliminaries
Let be a connected graph, where and be the set of vertices and edges of , respectively. For any vertex , its degree is denoted by and defined as the number of edges connecting it. The joining of two vertices formed an edge denoted by . Gutman and Trinajstic [5] introduced Zagreb indices and to calculate total electron energy of hydrocarbons. The Zagreb indices and are defined as
Ashrafi et al. [24] defined Zagreb coincides such as and of the Zagreb indices. The Zagreb coindices are defined as
Furtula and Gutman [16] defined forgotten index (-index), and its mathematical form is given by
Nilanjan et al. [41] introduced the forgotten coindex (-coindex) for -index with mathematical formulation as
Let be a simple connected graph; then, its derived graphs are defined as follows.(i) is a graph formed by inserting a new vertex in each edge of (ii) is a graph obtained from by adding an edge between the adjacent vertices of (iii) is a graph formed from by adding an edge between the pairs of new vertices which are on the adjacent edges of (iv) is formed by performing both operations of and on
Suppose that and are two connected graphs; then, their derived sum graph (D-sum graphs) is denoted by and defined with vertex set , and edge set is defined as the vertices and of , where are joined iff.(i) and (ii) and
Where presents the is an edge in [26]. For the -sum graphs, refer Figures 1 and 2.
(a)
(b)
(c)
(d)
(e)
3. Main Results
In this section, we discuss main results related to -coindex for -sum graphs. Let be a graph of its order, and the size is denoted by and , respectively. Let be the complement of , and the edge set for is given by . Further assumed some sums by , , , , and are given as follows.(i)(ii)(iii)(iv)(v)
Theorem 1. Let be a -sum graph; then, its -coindex is
Proof. Using equation (4), we haveBy substituting the values in equation (6), we get the required result.
Theorem 2. Let be a -sum graph; then, its -coindex is
Proof. Using equation (4), we haveUsing equation (7),By substituting the values in equation (10), we get the required result.
Theorem 3. Let be a -sum graph; then, -coindex is
Proof. Using equation (4), we haveConsequently,Using equation (8), we directly haveBy substituting the values in equation (13), we get the required result.
Theorem 4. Let be a -sum graph; then, its -coindex is
Proof. It follows from Theorems 2 and 3.
4. Conclusion
In this study, we have computed forgotten coindex of -sum graphs such as , , , and in their general forms. If and of order , then(i)In this study, we computed the exact values of forgotten coindex of -sum graphs such as , where in the form of forgotten index, first Zagreb indices, and coincides their basic graphs and .(ii)We illustrated Theorems 1–4 with the help of some particular path graphs such as , , , and for . Forgotten coindex of -sum graphs from the paths is also computed as applications of the obtained results.(iii)Table 1 and Figure 3 present that forgotten coindex of is dominant than forgotten coindex of , , and .
Open problem 1. Investigate the existence of the distance-based coindices for the -sum graphs.
Data Availability
The data used to support the findings of this study are included within the article and are available from the corresponding author upon request.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
The authors are very thankful for the referees’ comments, which helped them to improve the results and presentation of the manuscript to a great extent. This study was partially supported by University of Tabuk, Tabuk, Saudi Arabia.