In crystallography and the theory of infinite vertex-transitive graphs, the '''coordination sequence''' of a vertex <math>v</math> is an integer sequence that counts how many vertices are at each possible distance from <math>v</math>. That is, it is a sequence <math display=block>n_0, n_1, n_2,\dots</math> where each <math>n_i</math> is the number of vertices that are <math>i</math> steps away from <math>v</math>. If the graph is vertex-transitive, then the sequence is an invariant of the graph that does not depend on the specific choice of <math>v</math>. 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.{{r|brunner|conslo}}
thumb|A square grid, shaded by distance from the central blue point. The number of grid points at distance exactly <math>i > 0</math> is <math>4i</math>, 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 <math>i</math>, there are <math>4i</math> grid points that are <math>i</math> steps away from the origin. Therefore, the coordination sequence of the square grid is the sequence <math display=block>1,4,8,12,16,20,\dots\ .</math> in which, except for the initial value of one, each number is a multiple of four.{{r|A008574}}
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{{r|conslo|okeefe}} and uniform tilings are known.{{r|gooslo}}{{r|shutovmaleev}}
The coordination sequences of periodic structures are known to be quasi-polynomial.{{r|nakamura}}{{r|Kopczyński}}
==References== <references>
<ref name=A008574>{{cite OEIS|A008574|mode=cs2}}</ref>
<ref name=brunner>{{citation | last = Brunner | first = G. O. | date = July 1979 | doi = 10.1016/0022-4596(79)90207-x | issue = 1 | journal = Journal of Solid State Chemistry | pages = 41–45 | title = The properties of coordination sequences and conclusions regarding the lowest possible density of zeolites | volume = 29| bibcode = 1979JSSCh..29...41B }}</ref>
<ref name=conslo>{{citation | last1 = Conway | first1 = J. H. | author1-link = John Horton Conway | last2 = Sloane | first2 = N. J. A. | date = November 1997 | doi = 10.1098/rspa.1997.0126 | issue = 1966 | journal = Proceedings of the Royal Society A | mr = 1480120 | pages = 2369–2389 | title = Low-dimensional lattices. VII. Coordination sequences | volume = 453| bibcode = 1997RSPSA.453.2369C | s2cid = 120323174 }}</ref>
<ref name=gooslo>{{citation | last1 = Goodman-Strauss | first1 = C. | last2 = Sloane | first2 = N. J. A. | author2-link = Neil Sloane | arxiv = 1803.08530 | date = January 2019 | doi = 10.1107/s2053273318014481 | issue = 1 | journal = Acta Crystallographica Section A | mr = 3896412 | pages = 121–134 | title = A coloring-book approach to finding coordination sequences | url = https://neilsloane.com/doc/Cairo_final.pdf | volume = 75 | pmid = 30575590 | s2cid = 4553572 | access-date = 2021-06-18 | archive-date = 2022-02-17 | archive-url = https://web.archive.org/web/20220217162848/http://neilsloane.com/doc/Cairo_final.pdf | url-status = dead }}</ref>
<ref name=okeefe>{{citation | last = O'Keeffe | first = M. | date = January 1995 | doi = 10.1524/zkri.1995.210.12.905 | issue = 12 | journal = Zeitschrift für Kristallographie – Crystalline Materials | title = Coordination sequences for lattices | volume = 210| pages = 905–908 | bibcode = 1995ZK....210..905O }}</ref>
<ref name=shutovmaleev>{{citation | last1 = Shutov | first1 = Anton | last2 = Maleev | first2 = Andrey | date = 2020 | doi = 10.1515/zkri-2020-0002 | journal = Zeitschrift für Kristallographie – Crystalline Materials | title = Coordination sequences for lattices | volume = 235 | pages = 157–166 }}</ref>
<ref name=nakamura>{{citation | last1 = Nakamura | first1 = Y. | last2 = Sakamoto | first2 = R. | last3 = Mase | first3 = T. | last4 = Nakagawa | first4 = J. | date = 2021 | doi = 10.1107/S2053273320016769 | journal = Acta Crystallogr. | title = Coordination sequences of crystals are of quasi-polynomial type | volume = A77 | issue = 2 | pages = 138–148 | pmid = 33646200 | pmc = 7941273 | bibcode = 2021AcCry..77..138N }}</ref>
<ref name=Kopczyński>{{citation | last = Kopczyński | first = Eryk | date = 2022 | doi = 10.1107/S2053273322000262 | journal = Acta Crystallogr. | title = Coordination sequences of periodic structures are rational via automata theory | volume = A78 | issue = 2 | pages = 155–157 | pmid = 35230271 | arxiv = 2307.15803 }}</ref>
</references>
Category:Crystallography Category:Infinite graphs Category:Integer sequences