Combinatorics of colorings of Dűrer graph, Dűrer solid in Melencolia I, their heptagonal analogues and antiprisms for all irreducible representations: chirality, dynamics, NMR spectra and enumerations

Abstract

The Dűrer Solid has an art and mathematical history ever since it appeared in Dűrer’s 1514 copper engraving, Melencolia I. The associated truncated triangular trapezohedron is an intriguing structure owing to the symmetry, chirality concepts and combinatorial properties. The heptahedral version corresponds to the generalized Petersen’s graph G(7,2). It is shown that the heptagonal antiprism exhibits interesting combinatorial and chirality properties. The heptagonal bipyramid boron complex has been studied recently, and thus the developed techniques were applied to the multiple-quantum NMR of this complex. These graphs and solids serve as representations in dynamic stereochemistry of fluxional molecules as well as in material and molecular science. We consider symmetry-driven combinatorial-group theory through generating functions for the face, edge and vertex colorings of the title graphs and solids for all irreducible representations of the respective symmetry groups. We demonstrate of our computational-combinatorics techniques with tables of combinatorial numbers for the various colorings of these graphs for all irreducible representations. The chirality of the colorings can be directly inferred from the combinatorial enumerations. Applications to the ¹⁰B and ¹¹B MQ-NMR and nuclear spin statistics of the heptagonal bipyramid complex are considered.

Graphical abstract