Enumerating tree-like polyphenyl isomers

Abstract

Enumeration of molecules is one of the fundamental problems in bioinformatics and plays an important role in drug discovery, experimental structure elucidation (e.g., by using NMR or mass spectrometry), molecular design and virtual library construction. We consider the enumeration of tree-like polyphenyls (C6nH4n+2). For this purpose, we define two generating functions T(x) and R(x) involving the numbers tn and rn of tree-like polyphenyls (TL-polyphenyls) and monosubstituted tree-like polyphenyls (MTL-polyphenyls), respectively. By characterizing the symmetry groups with respect to TL-polyphenyls and MTL-polyphenyls, we establish two functional equations for these two generating functions. This yields for the first time an efficient recursion formula for calculating the numbers tn and rn. The two functional equations are also the fundamentals for analyzing their asymptotic behaviors, from which we derive the precise asymptotic values for both rn and tn. The resulting asymptotic values are shown to fit well to the numerical results obtained by using our recursion formula. Finally, we give an explicit enumerating expression for TL-polyphenyls of a particular type: the linear polyphenyls.