On the Role of Half-Catalan Numbers and Pathwidth in Hyperbranched Polymers Synthesized by ABmStep Polymerization

Topological features of polymer chains have been used as the key controlling mechanism for the physicochemical properties of hyperbranched polymers (HPs) and, therefore, provide a significant impetus to determine their branching characteristics. Single monomer methodology (SMM) involving ABm step polymerization has been one of the routes to synthesize both compact and segmented HPs. Here, we explore Catalan and half-Catalan numbers in the context of ABm step polymerization to deduce the structural information of HPs. Our approach harnesses the concepts of combinatorics and graph theory to calculate the exact numbers of isomeric, branched and linear, structures of polymer chains. We also demonstrate that the extent of branching of a polymer chain can be measured via pathwidth and establish its bounds as a function of its length. We believe that our findings can be leveraged to design and control the architecture of HPs synthesized via SMM or any other chemistries in a straightforward manner.