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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 ∈ S , R , Q , T and the connected graph H , the derived graphs D H 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 H 1 . 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.

Distance-based dimensions provide the foreground for the identification of chemical compounds that are chemically and structurally different but show similarity in different reactions. The reason behind this similarity is the occurrence of a set S of atoms and their same relative distances to some ordered set T of atoms in both compounds. In this article, the aforementioned problem is considered as a test case for characterising the (molecular) graphs bearing the fractional metric dimension (FMD) as 1. For the illustration of the theoretical development, it is shown that the FMD of path graph is unity. Moreover, we evaluated the extremal values of fractional metric dimension of a tetrahedral diamond lattice.

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 F -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 F -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.