p-Periodic nets can be derived from a voltage graph G with voltages in Zp, the free abelian group of rank p, if the cyclomatic number gamma of G is larger than p. Equivalently, one may describe a net by providing a set of (gamma - 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 subdeterminant 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.
Totally unimodular nets / J.G. Eon, D.M. Proserpio, V.A. Blatov. - In: ACTA CRYSTALLOGRAPHICA. SECTION A, FOUNDATIONS OF CRYSTALLOGRAPHY. - ISSN 0108-7673. - 68:2(2012), pp. 286-294. [10.1107/S0108767312000062]
Totally unimodular nets
D.M. ProserpioSecondo
;
2012
Abstract
p-Periodic nets can be derived from a voltage graph G with voltages in Zp, the free abelian group of rank p, if the cyclomatic number gamma of G is larger than p. Equivalently, one may describe a net by providing a set of (gamma - 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 subdeterminant 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.File | Dimensione | Formato | |
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