Computational Invariant of Chemical Structures and their ApplicationsView this Special Issue
Forgotten Index of Generalized Operations on Graphs
In theoretical chemistry, several distance-based, degree-based, and counting polynomial-related topological indices (TIs) are used to investigate the different chemical and structural properties of the molecular graphs. Furtula and Gutman redefined the -index as the sum of cubes of degrees of the vertices of the molecular graphs to study the different properties of their structure-dependency. In this paper, we compute -index of generalized sum graphs in terms of various TIs of their factor graphs, where generalized sum graphs are obtained by using four generalized subdivision-related operations and the strong product of graphs. We have analyzed our results through the numerical tables and the graphical presentations for the particular generalized sum graphs constructed with the help of path (alkane) graphs.
Throughout the paper, we consider a simple and undirected graph with vertex-set and edge-set , where and . A molecular graph is a connected and undirected graph in which atoms are presented by vertices, and chemical bonds between these atoms are shown by edges (see Figure 1). For a finite set of graphs and set of real numbers , the function defined by is called a degree-based topological index (TI), where the graph belongs to and is a degree-function from the vertex-set of to the degree-set of its vertices. It is important to know that if and only if is isomorphic to . For more details, see [1, 2].
Graph-theoretic modeling of the molecular graphs plays a fundamental part in the analysis of the quantitative structures activity/property relationships (QSAR/QSPR). In chemistry, the study of structural relationships is used to characterize the various physicochemical properties of organic molecules such as surface tension, density, melting, freezing point, solubility, heat of evaporation, and heat of formation . In last two decades, many TIs are introduced, but degree-based TIs got much more attention of the researchers, see the latest survey . In 1947, Winer introduced the first distance-based TI to compute the boiling point of paraffin . Also, we refer .
In molecular graph theory, the different operations on a graph perform a fundamental role in the formation of different new classes of graphs. Yan et al.  introduced the four operations , , , and on a graph and computed the Wiener indices for the graphs , where . Eliasi and Taeri  defined the -sum graphs using the Cartesian product on graphs and , where and are any two simple and connected graphs. They also computed the Wiener indices for these -sum graphs. Furthermore, Deng et al.  calculated the and Zagreb indices, Imran and Shehnaz  computed the -index, Liu et al.  computed the first general Zagreb, Chu et al.  calculated the bounds of first general Zagreb index and general Randic index, and Sarala et al.  computed the -index for these -sum graphs under the Cartesian and strong products. We also refer [14–20].
Recently, for , Liu et al.  defined the generalized -sum (-sum) graphs using Cartesian product and computed their Zagreb indices. Moreover, Awais et al. computed the -index , hyper-Zagreb , and FGZ index  for these generalized -sum graphs. Recently, Javaid et al.  computed first and second Zagreb indices for the generalized -sum graphs under the strong product.
In this paper, we compute -index of generalized sum graphs in terms of various TIs of their factor graphs, where generalized sum graphs are obtained by using four generalized subdivision-related operations and the strong product of graphs. We have analyzed our results through the numerical tables and the graphical presentations for the particular generalized sum graphs constructed with the help of path (alkane) graphs. The remaining paper is managed as follows: Section 2 consists of elementary definitions, Section 3 includes main results, and Section 4 covers the application and conclusion.
2.1. Degree-Based Topological Indices
In 1972, Trainajsi and Gutman defined first and second Zagreb indices that are utilized to find the -electron energy of molecular graphs . Let be any graph (molecular structure), then the first and second Zagreb indices are defined as and .
They also verified that the different predictive abilities of -index and first Zagreb index are same. In particular, both the indices yield the values of correlation coefficients for entropy and acentric factor more than 0.95. For more results on its mathematical properties and chemical applications, see .
2.2. Four Generalized Operations
Let be a (molecular) graph, then for the integral value , the graphs ( under the four generalized operations () on are defined as follows :(i)-subdivision graph: we add new vertices in every edge of and obtain the new -subdivision graph (ii)-semitotal point graph: the graph -semitotal point is defined from the graph by joining the vertices of which were adjacent in (iii)-semitotal line graph: the -semitotal line graph is defined from the graph by joining the newly added vertices for each incident pair of edges of (iv)-total point graph: the -total point graph is defined from the graph by applying both the operations and , respectively. For more details, see Figures 2–4.
2.3. Generalized Sum Operation for Strong Product
Let and be two graphs, presents generalized operations and is obtained after applying on having edge-set and node-set . The generalized -sum graphs under the operation of strong product is a graph having a vertex-set:such that two vertices and of are adjacent iff and or and or and , where is natural number. Furthermore, the generalized -sum graphs contain copies of graphs that are labeled with the vertices of . For more explanation, see Figures 5–7.
This section presents the main results.
Theorem 1. Let and be two connected graphs, then
Proof. Let is the degree of the vertex in , thenSince and ), thereforeWe know that .Since in this case , soWe arrive at our desired result.
Theorem 2. Let and be two connected graphs, then
Proof. Let is the degree of the vertex in , thenWe arrive at our desired result.
Theorem 3. Let and be two connected graphs, then
Proof. Let is the degree of the vertex in , thenConsider and occurs times. Thus,Letas and occurs two times. Therefore,Now, we split sum in two parts, and , where . Suppose that , where covers the edges of which are in the same edges of and of in two different adjacent edges of .In , the coefficient ofTherefore,For the coefficient of , let with and . As , we have either = or or = or . So, is adjacent to all those vertices in which are adjacent to and . So, the number of such is . Therefore,So,