Eon Jean-Guillaume, Proserpio Davide M, Blatov Vladislav A
Instituto de Química, Universidade Federal do Rio de Janeiro, Rio de Janeiro, Brazil.
Acta Crystallogr A. 2012 Mar;68(Pt 2):286-94. doi: 10.1107/S0108767312000062. Epub 2012 Feb 16.
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.
如果电压图(G)的圈数(\gamma)大于(p)，则(p)周期网可以从电压在(Z(p))（秩为(p)的自由阿贝尔群）中的电压图(G)导出。等价地，可以通过提供一组(G)的((\gamma - p))个循环向量来描述一个网，这些循环向量构成(G)的循环空间中净电压为零的子空间的一个基。设(M)是在(G)的(1 -)链空间的边基中表示的这个基的矩阵。当(M)的每个子行列式都属于集合({-1, 0, 1})时，一个网称为完全幺模的。从某个有限图只能导出有限个完全幺模网。结果表明，完全幺模网在边格删除操作下是稳定的，这使得它们与最小网具有可比性。给出了一个用于完全确定从某个有限图导出的完全幺模网的算法。作为一个应用，给出了从圈数为(3)和(4)且无桥的图导出的完全幺模网的完整列表。结果表明，许多完全幺模网经常出现在晶体结构中。
