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


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.


Nets Crystal structures Quotient graphs Space groups Coordination sequences 



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


  1. 1.
    Delgado-Friedrichs O, Foster MD, O’Keeffe M, Proserpio DM, Treacy MMJ, Yaghi OM (2005) What do we know about three-periodic nets? J Solid State Chem 178:2533–2554CrossRefGoogle Scholar
  2. 2.
    Chung SJ, Hahn Th, Klee WE (1984) Nomenclature and generation of 3-periodic nets—the vector method. Acta Cryst A40:42–50Google Scholar
  3. 3.
    Delgado-Friedrichs O, O’Keeffe M (2003) Identification and symmetry computation for crystal nets. Acta Cryst A59:351–360Google Scholar
  4. 4.
    Eon J-G (2004) Topological density of nets: a direct calculation. Acta Cryst A60:7–18Google Scholar
  5. 5.
    Blatov VA (2006) Multipurpose crystallochemical analysis with the program package TOPOS. IUCr CompComm Newsl 7:4–38Google Scholar
  6. 6.
    Klein H-J (1996) Systematic generation of models for crystal structures. Math Model Sci Comput 6:325–330Google Scholar
  7. 7.
    Koch E, Fischer W (1995) Sphere packings with three contacts per sphere and the problem of the least dense sphere packing. Z Kristallogr 210:407–414CrossRefGoogle Scholar
  8. 8.
    Treacy MMJ, Randall KH, Rao S, Perry JA, Chadi DJ (1997) Enumeration of periodic tetrahedral frameworks. Z Kristallogr 212:768–791CrossRefGoogle Scholar
  9. 9.
    Eon J-G (2002) Algebraic determination of generating functions for coordination sequences in crystal structures. Acta Cryst A58:47–53Google Scholar
  10. 10.
    Klee WE (2004) Crystallographic nets and their quotient graphs. Cryst Res Technol 39:959–968CrossRefGoogle Scholar
  11. 11.
    Gross JL, Tucker ThW (2000) Topological graph theory. Dover, MineolaGoogle Scholar
  12. 12.
    Eon J-G (2011) Euclidian embeddings of periodic nets: definition of a topologically induced, complete set of geometric descriptors for crystal structures. Acta Cryst A67:68–86Google Scholar
  13. 13.
    OKeeffe M, Peskov MA, Ramsden SJ, Yaghi OM (2008) The Reticular Chemistry Structure Resource (RCSR) database of, and symbols for, crystal nets. Acc Chem Res 41:1782–1789CrossRefGoogle Scholar
  14. 14.
    Eon J-G (2007) Infinite geodesic paths and fibers, new topological invariants in periodic graphs. Acta Cryst A63:53–65Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2012

Authors and Affiliations

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

Personalised recommendations