Coordination sequence

In crystallography and the theory of infinite vertex-transitive graphs, the coordination sequence of a vertex v is an integer sequence that counts how many vertices are at each possible distance from v. That is, it is a sequence n0,n1,n2,… where each ni is the number of vertices that are i steps away from v. If the graph is vertex-transitive, then the sequence is an invariant of the graph that does not depend on the specific choice of v. Coordination sequences can also be defined for sphere packings, by using either the contact graph of the spheres or the Delaunay triangulation of their centers, but these two choices may give rise to different sequences.[1][2]

A square grid, shaded by distance from the central blue point. The number of grid points at distance exactly i>0 is 4i, so the coordination sequence of the grid is the sequence of multiples of four, modified to start with one instead of zero.

As an example, in a square grid, for each positive integer i, there are 4i grid points that are i steps away from the origin. Therefore, the coordination sequence of the square grid is the sequence 1,4,8,12,16,20,… . in which, except for the initial value of one, each number is a multiple of four.[3]

The concept was proposed by Georg O. Brunner and Fritz Laves and later developed by Michael O'Keefe. The coordination sequences of many low-dimensional lattices[2][4] and uniform tilings are known.[5][6]

The coordination sequences of periodic structures are known to be quasi-polynomial.[7][8]

## References

1. ↑ Brunner, G. O. (July 1979), "The properties of coordination sequences and conclusions regarding the lowest possible density of zeolites", Journal of Solid State Chemistry 29 (1): 41–45, doi:10.1016/0022-4596(79)90207-x, Bibcode: 1979JSSCh..29...41B
2. ↑ 2.0 2.1 "Low-dimensional lattices. VII. Coordination sequences", Proceedings of the Royal Society A 453 (1966): 2369–2389, November 1997, doi:10.1098/rspa.1997.0126, Bibcode: 1997RSPSA.453.2369C
3. ↑ Sloane, N. J. A., ed. "Sequence A008574". OEIS Foundation. https://oeis.org/A008574.
4. ↑ O'Keeffe, M. (January 1995), "Coordination sequences for lattices", Zeitschrift für Kristallographie – Crystalline Materials 210 (12): 905–908, doi:10.1524/zkri.1995.210.12.905, Bibcode: 1995ZK....210..905O
5. ↑ Goodman-Strauss, C.; Sloane, N. J. A. (January 2019), "A coloring-book approach to finding coordination sequences", Acta Crystallographica Section A 75 (1): 121–134, doi:10.1107/s2053273318014481, PMID 30575590, https://neilsloane.com/doc/Cairo_final.pdf, retrieved 2021-06-18
6. ↑ Shutov, Anton; Maleev, Andrey (2020), "Coordination sequences for lattices", Zeitschrift für Kristallographie – Crystalline Materials 235: 157–166, doi:10.1515/zkri-2020-0002
7. ↑ Nakamura, Y.; Sakamoto, R.; Mase, T.; Nakagawa, J. (2021), "Coordination sequences of crystals are of quasi-polynomial type", Acta Crystallogr. A77: 138–148, doi:10.1107/S2053273320016769
8. ↑ Kopczyński, Eryk (2022), "Coordination sequences of periodic structures are rational via automata theory", Acta Crystallogr. A78: 155–157, doi:10.1107/S2053273322000262

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