Totally unimodular nets

Acta Crystallographica. Section A, Foundations of Crystallography
Jean-Guillaume EonVladislav A Blatov

Abstract

p-Periodic nets can be derived from a voltage graph G with voltages in Z(p), the free abelian group of rank p, if the cyclomatic number γ of G is larger than p. Equivalently, one may describe a net by providing a set of (γ - p) cycle vectors of G forming a basis of the subspace of the cycle space of G with zero net voltage. Let M be the matrix of this basis expressed in the edge basis of the 1-chain space of G. A net is called totally unimodular whenever every sub-determinant of M belongs to the set {-1, 0, 1}. Only a finite set of totally unimodular nets can be derived from some finite graph. It is shown that totally unimodular nets are stable under the operation of edge-lattice deletion in a sense that makes them comparable to minimal nets. An algorithm for the complete determination of totally unimodular nets derived from some finite graph is presented. As an application, the full list of totally unimodular nets derived from graphs of cyclomatic numbers 3 and 4, without bridges, is given. It is shown that many totally unimodular nets frequently occur in crystal structures.

References

May 31, 2002·Acta Crystallographica. Section B, Structural Science·Frank H Allen
Jul 2, 2003·Acta Crystallographica. Section A, Foundations of Crystallography·Olaf Delgado-Friedrichs, Michael O'Keeffe
Oct 28, 2004·Acta Crystallographica. Section A, Foundations of Crystallography·Charlotte BonneauOmar M Yaghi
Mar 16, 2005·Accounts of Chemical Research·Nathan W OckwigOmar M Yaghi
Dec 21, 2006·Acta Crystallographica. Section A, Foundations of Crystallography·Jean Guillaume Eon
Jun 16, 2007·Acta Crystallographica. Section A, Foundations of Crystallography·Vladislav A Blatov
Oct 7, 2008·Accounts of Chemical Research·Michael O'KeeffeOmar M Yaghi
Feb 20, 2009·Acta Crystallographica. Section A, Foundations of Crystallography·S J RamsdenS T Hyde
Apr 8, 2009·Acta Crystallographica. Section A, Foundations of Crystallography·Vladislav A Blatov, Davide M Proserpio
Dec 22, 2010·Acta Crystallographica. Section A, Foundations of Crystallography·Jean-Guillaume Eon
Apr 14, 2011·Acta Crystallographica. Section A, Foundations of Crystallography·Montauban Moreira de Oliveira, Jean Guillaume Eon

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Citations

Sep 6, 2018·Acta Crystallographica. Section A, Foundations and Advances·Montauban Moreira de OliveiraJean Guillaume Eon

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