Nanotubes: number of Kekulé structures and aromaticity

Use of Fibonacci numbers in lipidomics - Enumerating various classes of fatty acids

In lipid biochemistry, a fundamental question is how the potential number of fatty acids increases with their chain length. Here, we show that it grows according to the famous Fibonacci numbers when cis/trans isomerism is neglected. Since the ratio of two consecutive Fibonacci numbers tends to the Golden section, 1.618, organisms can increase fatty acid variability approximately by that factor per carbon atom invested. Moreover, we show that, under consideration of cis/trans isomerism and/or of modification by hydroxy and/or oxo groups, diversity can be described by generalized Fibonacci numbers (e.g. Pell numbers). For the sake of easy comprehension, we deliberately build the proof on the recursive definitions of these number series. Our results should be of interest for mass spectrometry, combinatorial chemistry, synthetic biology, patent applications, use of fatty acids as biomarkers and the theory of evolution. The recursive definition of Fibonacci numbers paves the way to construct all structural formulas of fatty acids in an automated way.

An Alternative Strategy for Count and Storage of Kekulé and Longer Range Resonance Valence Bond Structures

In this paper, we suggest a simple representation for notation of Kekulé valence bond (KVB) structures and longer range resonance valence bond (RVB) structures, which is called "the adjacency bonding array". In this representation, only an N component one-dimensional array is needed for inscribing each KVB or longer range RVB structure for an N-carbon system. Based on the adjacency bonding arrays, we develop very efficient algorithms for the systematic search of KVB and RVB structures as well as evaluation of the basis set overlap and Hamiltonian matrices.