Vertex and edge resolvability of some drug structures

Abstract

Distance-based parameters are crucial in studying drug structures as they help in uniquely identifying and characterizing the molecular structure through the analysis of graph representations, thereby aiding in the understanding of molecular properties and behaviors. A subset of vertex set of a graph is said to be vertex resolving set if no two vertices have same representation w.r.t. the resolving set. the minimum cardinality of the vertex resolving set is called metric dimension. Similarly, if the vertex subset of graph is responsible for uniquely identifying the edges of the graph then it is called edge resolving set, and the minimum cardinality of edge resolving set is called the edge metric dimension. In pharmaceutical research, resolving sets are especially used to identify patterns shared by multiple medications. Medical drugs have been an integral part of human culture since the beginning of time. In this study, we investigated the metric and edge metric dimensions of some significant drug structures namely, Salicylic Acid $\left(S{A}_{16}\right)$, Diazepam $\left(D{I}_{33}\right)$, Lidocaine $\left(L{I}_{39}\right)$ and Ibuprofen $\left(I{B}_{39}\right)$. According to our findings, we observed that we have a relation for $\left(D{I}_{33}\right)$, $dim\left(D{I}_{16}\right)\ge edim\left(D{I}_{16}\right)$. The metric and edge metric dimension are same for $\left(S{A}_{16}\right)$, $\left(L{I}_{39}\right)$ and $\left(I{B}_{33}\right)$ but if we compare two different molecules form them, their metric (edge) dimension is not identical. The fact that their metric dimensions are not identical makes it more efficient and effective for molecular structure identification.