Forgotten Coindex for the Derived Sum Graphs under Cartesian Product

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 $H$ is defined as the sum of the degrees of all the possible pairs of nonadjacent vertices in $H$ . For $D\in \left\{S,R,Q,T\right\}$ and the connected graph $H$ , the derived graphs $D\left(H\right)$ are obtained by applying the operations $S$ (subdivided), $R$ (triangle parallel), $Q$ (line superposition), and $T$ (total graph), respectively. Moreover, a derived sum graph ( $D$ -sum graph) is obtained by the Cartesian product of the graph $H_2$ with the graph $D\left(H_1\right)$ . In this study, we compute forgotten coindex of the $D$ -sum graphs $H_{1+S}{H}_2$ ( $S$ -sum), $H_{1+R}{H}_2$ ( $R$ -sum), $H_{1+Q}{H}_2$ ( $Q$ -sum), and $H_{1+T}{H}_2$ ( $T$ -sum) in the form of various indices and coindices of the factor graphs $H_1$ and $H_2$ . At the end, we have analyzed our results using numerical tables and graphical behaviour for some particular $D$ -sum graphs.