A massive of early drug tests implies that there exist strong inner relationships between the bio-medical and pharmacology characteristics of drugs and their molecular structures. The forgotten topological index was defined to be used in the analysis of drug molecular structures, which is quite helpful for pharmaceutical and medical scientists to grasp the biological and chemical characteristics of new drugs. Such tricks are popularly employed in developing countries where enough money is lacked to afford the relevant chemical reagents and equipment. In our article, by means of drug molecular structure analysis and edge dividing technology, we present the forgotten topological index of several widely used chemical structures which often appear in drug molecular graphs.

Numerous studies indicate that there is strong inherent relationship between the chemical characteristics of chemical compounds and drugs (e.g., boiling point and melting point) and their molecular structures. Topological indices defined on these chemical molecular structures can help researchers better understand the physical features, chemical reactivity, and biological activity. Thus, the study of the topological indices on chemical structure of chemical materials and drugs can make up for lack of chemical experiments and can provide a theoretical basis for the manufacturing of drugs and chemical materials. In this paper, we focus on the family of smart polymer which is widely used in anticancer drugs manufacturing. Several topological indices are determined in view of edge dividing methods, and these results remedy the lack of chemical and medicine experiments thus providing the theoretical basis for pharmaceutical engineering.

Topological indices are graph invariants computed by the distance or degree of vertices of the molecular graph. In chemical graph theory, topological indices have been successfully used in describing the structures and predicting certain physicochemical properties of chemical compounds. In this paper, we propose a definition of generalized bridge molecular graphs that can model more kinds of long chain polymerization products than the bridge molecular graphs, and provide some results of the edge versions of atom-bond connectivity ( $AB{C}_e$ ) and geometric arithmetic ( $G{A}_e$ ) indices for some generalized bridge molecular graphs, which have regular, periodic and symmetrical structures. The results of this paper offer promising prospects in the applications for chemical and material engineering, especially in chemical industry research.

The Zagreb indices have been introduced more than forty years ago by I. Gutman, and N. Trinajstić and are defined as and in which degree of a vertex u denoted by d_u. In the present study, we compute these Zagreb topological indices for a type of Benzenoid Systems “jagged-rectangle”.

It is found from the earlier studies that the structure-dependency of totalπ-electron energyEπheavily relies on the sum of squares of the vertex degrees of the molecular graph. Hence, it provides a measure of the branching of the carbon-atom skeleton. In recent years, the sum of squares of the vertex degrees of the molecular graph has been defined as forgotten topological index which reflects the structure-dependency of totalπ-electron energyEπand measures the physical-chemical properties of molecular structures. In this paper, in order to research the structure-dependency of totalπ-electron energyEπ, we present the forgotten topological index of some important molecular structures from mathematical standpoint. The formulations we obtained here use the approach of edge set dividing, and the conclusions can be applied in physics, chemical, material, and pharmaceutical engineering.

A graph invariant is a numerical value that depicts the structural properties of an entire graph. The Wiener index is the oldest distance based graph invariant which is defined as the sum of distances between all unordered pair of vertices of the graph G. In this paper we use the method of edge cut to compute the Wiener index and Wiener polarity index of linear (L_n) and zig-zag (Z_n) polyomino chains of 4-cycle. Also, we introduce another polyomino chain of 4k-cycle Z_mn and calculate its Wiener index and Wiener polarity index for m = 3 and m = 4. Finally, the graphical representation is given to analyze the behavior of these indices.