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Structural Chemistry

, Volume 23, Issue 4, pp 987–996 | Cite as

From symmetry-labeled quotient graphs of crystal nets to coordination sequences

Algebraic tools for a combinatorial analysis of crystal structures
  • Jean-Guillaume Eon
Original Research

Abstract

The combinatorial topology of crystal structures may be described by finite graphs, called symmetry-labeled quotient graphs or voltage graphs, with edges labeled by symmetry operations from their space group. These symmetry operations themselves generate a space group which is generally a non-trivial subgroup of the crystal space group. The method is an extension of the so-called vector method, where translation symmetries are used as vector labels (voltages) for the edges of the graph. Non-translational symmetry operations may be used as voltages if they act freely on the net underlying the crystal structure. This extension provides a significant reduction of the size of the quotient graph. A few uninodal and binodal nets are examined as illustrations. In particular, various uninodal nets appear to be isomorphic to Cayley color graphs of space group. As an application, the full coordination sequence of the diamond net is determined.

Keywords

Nets Crystal structures Quotient graphs Space groups Coordination sequences 

Notes

Acknowledgments

The author thanks CNPq (Conselho Nacional de Desenvolvimento e Pesquisa) of Brazil for support during the preparation of this study.

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Copyright information

© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  1. 1.Instituto de QuímicaUniversidade Federal do Rio de JaneiroRio de JaneiroBrazil

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