26:
1506: ≡ 1 (mod 8) that are highly symmetric and self-dual, generalizing a natural embedding of the Paley graph of order 9 as a 3×3 square grid on a torus. However the genus of White's embeddings is higher by approximately a factor of three than Mohar's conjectured bound.
1278:
688:
968:
387:
1483:
1151:
817:
1384:
1394:
is a square, and questions whether such a bound might hold more generally. Specifically, Mohar conjectures that the Paley graphs of square order can be embedded into surfaces with genus
861:
766:
1610:
561:
1772:
Cramer, Kevin; Krebs, Mike; Shabazi, Nicole; Shaheen, Anthony; Voskanian, Edward (2016). "The isoperimetric and
Kazhdan constants associated to a Paley graph".
893:
293:
1762:
For obtaining the spectrum from strong regularity, see
Theorem 9.1.3, p. 116. For the connection to Gauss sums, see Section 9.8.5 Cyclotomy, pp. 138–140.
2073:
1747:
121:
994:) the same as for random graphs, and large sets of vertices have approximately the same number of edges as they would in random graphs.
1400:
1706:
2083:
1864:
875:
1273:{\displaystyle A=\left\{(a,b)\in \mathbf {F} _{q}\times \mathbf {F} _{q}\ :\ b-a\in (\mathbf {F} _{q}^{\times })^{2}\right\}.}
1654:
1390:
conjectures that the minimum genus of a surface into which a Paley graph can be embedded is near this bound in the case that
990:: the number of times each possible constant-order graph occurs as a subgraph of a Paley graph is (in the limit for large
771:
1054:
1989:
Baker, R. D.; Ebert, G. L.; Hemmeter, J.; Woldar, A. J. (1996). "Maximal cliques in the Paley graph of square order".
1307:
1322:
could be embedded so that all its faces are triangles, we could calculate the genus of the resulting surface via the
1329:
522:: the complement of any Paley graph is isomorphic to it. One isomorphism is via the mapping that takes a vertex
2010:
Broere, I.; Döman, D.; Ridley, J. N. (1988). "The clique numbers and chromatic numbers of certain Paley graphs".
1284:
494:
Thus, in the Paley graph, we form a vertex for each of the integers in the range , and connect each such integer
219:
1306:
The six neighbors of each vertex in the Paley graph of order 13 are connected in a cycle; that is, the graph is
1318:, in which every face is a triangle and every triangle is a face. More generally, if any Paley graph of order
1311:
519:
102:
165:
of quadratic residues, and have interesting properties that make them useful in graph theory more generally.
2078:
81:
2068:
822:
727:
552:
254:(a prime congruent to 1 mod 4) or an even power of an odd non-Pythagorean prime. This choice of
94:
222:
with a property previously known to be held only by random tournaments: in a Paley digraph, every small
46:
1693:. Ergebnisse der Mathematik und ihrer Grenzgebiete. Vol. 18. Berlin: Springer-Verlag. p. 10.
1323:
999:
1895:"Construction of rank two reflexive sheaves with similar properties to the Horrocks–Mumford bundle"
864:
697:
and self-complementary. The strongly regular graphs with parameters of this form (for an arbitrary
64:
2051:
1930:
1295:
683:{\displaystyle srg\left(q,{\tfrac {1}{2}}(q-1),{\tfrac {1}{4}}(q-5),{\tfrac {1}{4}}(q-1)\right).}
173:
1605:
197:
1743:
1702:
1291:
977:
710:
251:
211:
154:
146:
2019:
1998:
1942:
1906:
1873:
1825:
1781:
1735:
1694:
1663:
1619:
1573:
1536:
706:
702:
150:
138:
98:
1956:
1793:
1757:
1716:
1689:
Brouwer, A. E.; Cohen, A. M.; Neumaier, A. (1989). "Conference matrices and Paley graphs".
1675:
1631:
1587:
1952:
1789:
1753:
1712:
1671:
1627:
1583:
1287:
because each pair of distinct vertices is linked by an arc in one and only one direction.
980:
718:
694:
177:
158:
1003:
1528:
1129:
1014:
963:{\displaystyle \displaystyle {\frac {q-{\sqrt {q}}}{4}}\leq i(G)\leq {\frac {q-1}{4}}.}
430:
207:
2002:
2062:
1878:
1859:
1816:
1811:
1645:
1601:
1523:
1033:
193:
169:
162:
39:
382:{\displaystyle E=\left\{\{a,b\}\ :\ a-b\in (\mathbf {F} _{q}^{\times })^{2}\right\}}
2039:
1649:
1018:
1007:
142:
2023:
480:
is just integer arithmetic modulo 13. The numbers with square roots mod 13 are:
2047:
1926:
1557:
1387:
1071:
239:
189:
130:
1785:
1032:
nor its complement contains a complete 4-vertex subgraph. It follows that the
1739:
1698:
1578:
1561:
1053:
Sasukara et al. (1993) use Paley graphs to generalize the construction of the
1807:
1667:
1540:
1730:
Brouwer, Andries E.; Haemers, Willem H. (2012). "9.1.2 The Paley graphs".
709:
of a conference graph, such as a Paley graph, can be used to construct a
180:
from quadratic residues. They were introduced as graphs independently by
1911:
1894:
705:, so the Paley graphs form an infinite family of conference graphs. The
192:
was interested in them for their self-complementarity properties, while
1829:
1623:
223:
218:(independently of Sachs, Erdős, and Rényi) as a way of constructing
25:
1947:
1315:
1290:
The Paley digraph leads to the construction of some antisymmetric
1042:
The Paley graph of order 101 is currently the largest known graph
1860:"On the number of complete subgraphs contained in certain graphs"
1478:{\displaystyle (q^{2}-13q+24)\left({\tfrac {1}{24}}+o(1)\right),}
717:, with zero on the diagaonal, that give a scalar multiple of the
405:, it is included under either ordering of its two elements. For,
1091:, has no square root of −1. Consequently, for each pair (
713:, and vice versa. These are matrices whose coefficients are
1652:(1971). "A constructive solution to a tournament problem".
1893:
Sasakura, Nobuo; Enta, Yoichi; Kagesawa, Masataka (1993).
1050:
nor its complement contains a complete 6-vertex subgraph.
1024:
The Paley graph of order 17 is the unique largest graph
421:), and −1 is a square, from which it follows that
1734:. Universitext. New York: Springer. pp. 114–115.
1441:
1334:
827:
776:
732:
646:
616:
586:
1970:
White, A. T. (2001). "Graphs of groups on surfaces".
1403:
1332:
1154:
897:
896:
825:
774:
730:
693:
This in fact follows from the fact that the graph is
564:
296:
1974:. Amsterdam: North-Holland Mathematics Studies 188.
888:of the Paley graph satisfies the following bounds:
867:or by using the theory of strongly regular graphs.
108:
90:
80:
63:
45:
35:
18:
1477:
1378:
1272:
962:
855:
812:{\displaystyle {\tfrac {1}{2}}(-1\pm {\sqrt {q}})}
811:
760:
682:
381:
1858:Evans, R. J.; Pulham, J. R.; Sheehan, J. (1981).
1611:Acta Mathematica Academiae Scientiarum Hungaricae
210:analogs of Paley graphs that yield antisymmetric
145:by connecting pairs of elements that differ by a
1814:; Wilson, R. M. (1989). "Quasi-random graphs".
226:of vertices is dominated by some other vertex.
1502:finds embeddings of the Paley graphs of order
1379:{\displaystyle {\tfrac {1}{24}}(q^{2}-13q+24)}
153:, which yield an infinite family of symmetric
149:. The Paley graphs form an infinite family of
1310:. Therefore, this graph can be embedded as a
1078:= 3 (mod 4). Thus, the finite field of order
490:±4 (square roots ±2 for +4, ±3 for −4).
215:
8:
487:±3 (square roots ±4 for +3, ±6 for −3)
484:±1 (square roots ±1 for +1, ±5 for −1)
320:
308:
1488:where the o(1) term can be any function of
141:constructed from the members of a suitable
1847:. Master's Thesis. University of Tübingen.
976:is prime, the associated Paley graph is a
271:, the element −1 has a square root.
185:
24:
1946:
1931:"Triangulations and the Hajós conjecture"
1910:
1877:
1577:
1440:
1411:
1402:
1352:
1333:
1331:
1256:
1246:
1241:
1236:
1205:
1200:
1190:
1185:
1153:
938:
907:
898:
895:
826:
824:
799:
775:
773:
731:
729:
645:
615:
585:
563:
368:
358:
353:
348:
295:
250:should either be an arbitrary power of a
258:implies that in the unique finite field
1552:
1550:
1515:
506: ± 3 (mod 13), and
15:
1499:
863:). They can be calculated using the
181:
7:
856:{\displaystyle {\tfrac {1}{2}}(q-1)}
761:{\displaystyle {\tfrac {1}{2}}(q-1)}
724:The eigenvalues of Paley graphs are
721:when multiplied by their transpose.
1935:Electronic Journal of Combinatorics
1845:Engineering Linear Layouts with SAT
1566:Publicationes Mathematicae Debrecen
510: ± 4 (mod 13).
502: ± 1 (mod 13),
455:) is the Paley graph of order
172:. They are closely related to the
1562:"Über selbstkomplementäre Graphen"
1526:(1933). "On orthogonal matrices".
1492:that goes to zero in the limit as
1124:, but not both, is a square. The
14:
1237:
1201:
1186:
1013:The Paley graph of order 13 has
998:The Paley graph of order 9 is a
349:
1865:Journal of Combinatorial Theory
1655:Canadian Mathematical Bulletin
1464:
1458:
1432:
1404:
1373:
1345:
1253:
1232:
1178:
1166:
932:
926:
850:
838:
806:
787:
755:
743:
669:
657:
639:
627:
609:
597:
365:
344:
122:Table of graphs and parameters
1:
2074:Parametric families of graphs
2024:10.1080/16073606.1988.9631945
2003:10.1016/S0378-3758(96)00006-7
1608:(1963). "Asymmetric graphs".
168:Paley graphs are named after
1991:J. Statist. Plann. Inference
1879:10.1016/0095-8956(81)90054-X
1039:(4, 4) = 18.
216:Graham & Spencer (1971)
161:tools to be applied to the
30:The Paley graph of order 13
2100:
1786:10.2140/involve.2016.9.293
1099:) of distinct elements of
768:(with multiplicity 1) and
214:. They were introduced by
200:studied their symmetries.
1899:Proc. Japan Acad., Ser. A
1740:10.1007/978-1-4614-1939-6
1699:10.1007/978-3-642-74341-2
1579:10.5486/PMD.1962.9.3-4.11
120:
23:
2012:Quaestiones Mathematicae
819:(both with multiplicity
186:Erdős & Rényi (1963)
2084:Strongly regular graphs
2052:"Genus of Paley graphs"
1972:Interactions and models
1691:Distance-regular graphs
1283:The Paley digraph is a
1055:Horrocks–Mumford bundle
1006:, and the graph of the
553:strongly regular graphs
1843:Wolz, Jessica (2018).
1668:10.4153/CMB-1971-007-1
1541:10.1002/sapm1933121311
1479:
1380:
1274:
964:
857:
813:
762:
684:
383:
246:= 1 (mod 4). That is,
1480:
1381:
1312:Whitney triangulation
1275:
965:
858:
814:
763:
685:
518:The Paley graphs are
384:
157:. Paley graphs allow
2038:Brouwer, Andries E.
1401:
1330:
1324:Euler characteristic
1152:
1000:locally linear graph
894:
876:isoperimetric number
823:
772:
728:
562:
294:
1912:10.3792/pjaa.69.144
1292:conference matrices
1251:
865:quadratic Gauss sum
555:, with parameters
363:
212:conference matrices
155:conference matrices
1830:10.1007/BF02125347
1624:10.1007/BF01895716
1496:goes to infinity.
1475:
1450:
1376:
1343:
1296:biplane geometries
1270:
1235:
1046:such that neither
1028:such that neither
960:
959:
853:
836:
809:
785:
758:
741:
680:
655:
625:
595:
541:is any nonresidue
520:self-complementary
498:to six neighbors:
379:
347:
174:Paley construction
103:Self-complementary
1812:Graham, Ronald L.
1749:978-1-4614-1938-9
1732:Spectra of graphs
1449:
1342:
1219:
1213:
986:Paley graphs are
954:
918:
912:
835:
804:
784:
740:
711:conference matrix
703:conference graphs
654:
624:
594:
551:Paley graphs are
401:} is included in
331:
325:
252:Pythagorean prime
178:Hadamard matrices
176:for constructing
151:conference graphs
147:quadratic residue
139:undirected graphs
127:
126:
2091:
2055:
2043:
2027:
2006:
1976:
1975:
1967:
1961:
1960:
1950:
1923:
1917:
1916:
1914:
1890:
1884:
1883:
1881:
1855:
1849:
1848:
1840:
1834:
1833:
1808:Chung, Fan R. K.
1804:
1798:
1797:
1769:
1763:
1761:
1727:
1721:
1720:
1686:
1680:
1679:
1642:
1636:
1635:
1618:(3–4): 295–315.
1598:
1592:
1591:
1581:
1554:
1545:
1544:
1535:(1–4): 311–320.
1520:
1484:
1482:
1481:
1476:
1471:
1467:
1451:
1442:
1416:
1415:
1385:
1383:
1382:
1377:
1357:
1356:
1344:
1335:
1279:
1277:
1276:
1271:
1266:
1262:
1261:
1260:
1250:
1245:
1240:
1217:
1211:
1210:
1209:
1204:
1195:
1194:
1189:
1132:with vertex set
993:
975:
969:
967:
966:
961:
955:
950:
939:
919:
914:
913:
908:
899:
887:
873:
862:
860:
859:
854:
837:
828:
818:
816:
815:
810:
805:
800:
786:
777:
767:
765:
764:
759:
742:
733:
716:
707:adjacency matrix
700:
689:
687:
686:
681:
676:
672:
656:
647:
626:
617:
596:
587:
547:
540:
536:
525:
471:= 13, the field
388:
386:
385:
380:
378:
374:
373:
372:
362:
357:
352:
329:
323:
99:Conference graph
95:Strongly regular
28:
16:
2099:
2098:
2094:
2093:
2092:
2090:
2089:
2088:
2059:
2058:
2046:
2037:
2034:
2009:
1988:
1985:
1983:Further reading
1980:
1979:
1969:
1968:
1964:
1925:
1924:
1920:
1892:
1891:
1887:
1857:
1856:
1852:
1842:
1841:
1837:
1806:
1805:
1801:
1771:
1770:
1766:
1750:
1729:
1728:
1724:
1709:
1688:
1687:
1683:
1644:
1643:
1639:
1600:
1599:
1595:
1556:
1555:
1548:
1524:Paley, R.E.A.C.
1522:
1521:
1517:
1512:
1439:
1435:
1407:
1399:
1398:
1348:
1328:
1327:
1304:
1252:
1199:
1184:
1165:
1161:
1150:
1149:
1144:
1107:
1090:
1064:
991:
981:circulant graph
973:
970:
940:
900:
892:
891:
878:
871:
821:
820:
770:
769:
726:
725:
719:identity matrix
714:
698:
578:
574:
560:
559:
542:
538:
527:
523:
516:
479:
465:
364:
307:
303:
292:
291:
286:
266:
232:
159:graph-theoretic
101:
97:
55:
31:
12:
11:
5:
2097:
2095:
2087:
2086:
2081:
2079:Regular graphs
2076:
2071:
2061:
2060:
2057:
2056:
2044:
2040:"Paley graphs"
2033:
2032:External links
2030:
2029:
2028:
2007:
1984:
1981:
1978:
1977:
1962:
1918:
1905:(5): 144–148.
1885:
1872:(3): 364–371.
1850:
1835:
1824:(4): 345–362.
1799:
1780:(2): 293–306.
1764:
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1722:
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1681:
1650:Spencer, J. H.
1637:
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1529:J. Math. Phys.
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1062:Paley digraphs
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1015:book thickness
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431:if and only if
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1817:Combinatorica
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1229:
1226:
1223:
1220:
1214:
1206:
1196:
1191:
1181:
1175:
1172:
1169:
1162:
1158:
1155:
1148:
1147:
1146:
1145:and arc set
1143:
1139:
1135:
1131:
1127:
1126:Paley digraph
1123:
1119:
1115:
1111:
1106:
1102:
1098:
1094:
1089:
1085:
1081:
1077:
1073:
1069:
1061:
1056:
1052:
1049:
1045:
1041:
1038:
1035:
1034:Ramsey number
1031:
1027:
1023:
1020:
1016:
1012:
1009:
1005:
1001:
997:
996:
995:
989:
984:
982:
979:
956:
951:
947:
944:
941:
935:
929:
923:
920:
915:
909:
904:
901:
889:
885:
881:
877:
868:
866:
847:
844:
841:
832:
829:
801:
796:
793:
790:
781:
778:
752:
749:
746:
737:
734:
722:
720:
712:
708:
704:
701:) are called
696:
677:
673:
666:
663:
660:
651:
648:
642:
636:
633:
630:
621:
618:
612:
606:
603:
600:
591:
588:
582:
579:
575:
571:
568:
565:
558:
557:
556:
554:
549:
546:
534:
530:
521:
513:
511:
509:
505:
501:
497:
489:
486:
483:
482:
481:
478:
474:
470:
462:
460:
458:
454:
450:
446:
441:
440:is a square.
439:
436: −
435:
432:
428:
425: −
424:
420:
417: −
416:
412:
409: −
408:
404:
400:
396:
375:
369:
359:
354:
341:
338:
335:
332:
326:
317:
314:
311:
304:
300:
297:
290:
289:
288:
285:
281:
277:
272:
270:
265:
261:
257:
253:
249:
245:
241:
237:
229:
227:
225:
221:
217:
213:
209:
205:
201:
199:
195:
191:
187:
183:
179:
175:
171:
170:Raymond Paley
166:
164:
163:number theory
160:
156:
152:
148:
144:
140:
136:
132:
123:
119:
115:
111:
107:
104:
100:
96:
93:
89:
85:
83:
79:
75:
71:
68:
66:
62:
58:
53:
50:
48:
44:
41:
40:Raymond Paley
38:
34:
27:
22:
17:
2048:Mohar, Bojan
2015:
2011:
1994:
1990:
1971:
1965:
1938:
1934:
1927:Mohar, Bojan
1921:
1902:
1898:
1888:
1869:
1868:. Series B.
1863:
1853:
1844:
1838:
1821:
1815:
1802:
1777:
1773:
1767:
1731:
1725:
1690:
1684:
1659:
1653:
1640:
1615:
1609:
1596:
1569:
1565:
1558:Sachs, Horst
1532:
1527:
1518:
1503:
1500:White (2001)
1498:
1493:
1489:
1487:
1391:
1319:
1305:
1289:
1282:
1141:
1137:
1133:
1125:
1121:
1117:
1113:
1109:
1104:
1100:
1096:
1092:
1087:
1083:
1079:
1075:
1067:
1065:
1047:
1043:
1036:
1029:
1025:
1019:queue number
1008:3-3 duoprism
1004:rook's graph
988:quasi-random
987:
985:
971:
883:
879:
869:
723:
692:
550:
544:
532:
528:
517:
507:
503:
499:
495:
493:
476:
472:
468:
466:
456:
452:
448:
444:
442:
437:
433:
429:is a square
426:
422:
418:
414:
410:
406:
402:
398:
394:
392:
283:
279:
275:
273:
268:
263:
259:
255:
247:
243:
235:
233:
203:
202:
182:Sachs (1962)
167:
143:finite field
135:Paley graphs
134:
128:
113:
73:
69:
56:
51:
1572:: 270–288.
1388:Bojan Mohar
1072:prime power
978:Hamiltonian
393:If a pair {
240:prime power
220:tournaments
131:mathematics
59:prime power
36:Named after
19:Paley graph
2063:Categories
1510:References
1285:tournament
1074:such that
514:Properties
242:such that
230:Definition
91:Properties
54:≡ 1 mod 4,
2018:: 91–93.
1997:: 33–38.
1662:: 45–48.
1606:Rényi, A.
1602:Erdős, P.
1418:−
1359:−
1248:×
1230:∈
1224:−
1197:×
1182:∈
1108:, either
945:−
936:≤
921:≤
905:−
845:−
797:±
791:−
750:−
664:−
634:−
604:−
413:= −(
360:×
342:∈
336:−
267:of order
2050:(2005).
1929:(2005).
1560:(1962).
537:, where
287:and let
274:Now let
208:directed
109:Notation
82:Diameter
47:Vertices
1957:2176532
1941:: N15.
1794:3470732
1774:Involve
1758:2882891
1717:1002568
1676:0292715
1632:0156334
1588:0151953
1128:is the
715:±1
463:Example
451:,
1955:
1792:
1756:
1746:
1715:
1705:
1674:
1630:
1586:
1218:
1212:
1017:4 and
330:
324:
224:subset
76:− 1)/4
1316:torus
1314:of a
1302:Genus
1070:be a
972:When
531:(mod
238:be a
198:Rényi
194:Erdős
190:Sachs
65:Edges
1744:ISBN
1703:ISBN
1294:and
1066:Let
1002:, a
543:mod
467:For
234:Let
206:are
196:and
184:and
137:are
2020:doi
1999:doi
1943:doi
1907:doi
1874:doi
1826:doi
1782:doi
1736:doi
1695:doi
1664:doi
1620:doi
1574:doi
1537:doi
1326:as
1116:or
870:If
526:to
129:In
112:QR(
2065::
2016:11
2014:.
1995:56
1993:.
1953:MR
1951:.
1939:12
1937:.
1933:.
1903:69
1901:.
1897:.
1870:30
1862:.
1820:.
1810:;
1790:MR
1788:.
1776:.
1754:MR
1752:.
1742:.
1713:MR
1711:.
1701:.
1672:MR
1670:.
1660:14
1658:.
1648:;
1628:MR
1626:.
1616:14
1614:.
1604:;
1584:MR
1582:.
1568:.
1564:.
1549:^
1533:12
1447:24
1430:24
1421:13
1386:.
1371:24
1362:13
1340:24
1298:.
1136:=
1120:−
1112:−
1082:,
1021:3.
983:.
548:.
529:xk
459:.
278:=
188:.
133:,
2054:.
2042:.
2026:.
2022::
2005:.
2001::
1959:.
1945::
1915:.
1909::
1882:.
1876::
1832:.
1828::
1822:9
1796:.
1784::
1778:9
1760:.
1738::
1719:.
1697::
1678:.
1666::
1634:.
1622::
1590:.
1576::
1570:9
1543:.
1539::
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1494:q
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1469:)
1465:)
1462:1
1459:(
1456:o
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1444:1
1437:(
1433:)
1427:+
1424:q
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1409:q
1405:(
1392:q
1374:)
1368:+
1365:q
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1268:.
1264:}
1258:2
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1233:(
1227:a
1221:b
1215::
1207:q
1202:F
1192:q
1187:F
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1176:b
1173:,
1170:a
1167:(
1163:{
1159:=
1156:A
1142:q
1138:F
1134:V
1122:a
1118:b
1114:b
1110:a
1105:q
1101:F
1097:b
1095:,
1093:a
1088:q
1084:F
1080:q
1076:q
1068:q
1057:.
1048:G
1044:G
1037:R
1030:G
1026:G
1010:.
992:q
974:q
957:.
952:4
948:1
942:q
933:)
930:G
927:(
924:i
916:4
910:q
902:q
886:)
884:G
882:(
880:i
872:q
851:)
848:1
842:q
839:(
833:2
830:1
807:)
802:q
794:1
788:(
782:2
779:1
756:)
753:1
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744:(
738:2
735:1
699:q
678:.
674:)
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667:1
661:q
658:(
652:4
649:1
643:,
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637:5
631:q
628:(
622:4
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613:,
610:)
607:1
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598:(
592:2
589:1
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580:q
576:(
572:g
569:r
566:s
545:q
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533:q
524:x
508:x
504:x
500:x
496:x
477:q
473:F
469:q
457:q
453:E
449:V
445:G
438:a
434:b
427:b
423:a
419:a
415:b
411:b
407:a
403:E
399:b
397:,
395:a
389:.
376:}
370:2
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355:q
350:F
345:(
339:b
333:a
327::
321:}
318:b
315:,
312:a
309:{
305:{
301:=
298:E
284:q
280:F
276:V
269:q
264:q
260:F
256:q
248:q
244:q
236:q
116:)
114:q
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74:q
72:(
70:q
57:q
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