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| {{infobox graph
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| | name = Paley graph
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| | image = [[Image:Paley13.svg|220px]]
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| | image_caption = The Paley graph of order 13
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| | namesake = [[Raymond Paley]]
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| | vertices = ''q'' ≡ 1 mod 4,<br>''q'' prime power
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| | edges = ''q''(''q'' − 1)/4
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| | automorphisms
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| | chromatic_number =
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| | chromatic_index =
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| | properties = [[Strongly regular graph|Strongly regular]]<br>[[Conference graph]]<br>[[Self-complementary graph|Self-complementary]]
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| | notation = QR(''q'')
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| }}
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| In [[mathematics]], and specifically [[graph theory]], '''Paley graphs''', named after [[Raymond Paley]], are [[Dense graph|dense]] undirected graphs constructed from the members of a suitable [[finite field]] by connecting pairs of elements that differ in a [[quadratic residue]]. The Paley graphs form an infinite family of [[conference graph]]s, which yield an infinite family of symmetric [[conference matrix|conference matrices]]. Paley graphs allow graph-theoretic tools to be applied to the [[number theory]] of quadratic residues, and have interesting properties that make them useful in graph theory more generally.
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| Paley graphs are closely related to the [[Paley construction]] for constructing [[Hadamard matrix|Hadamard matrices]] from quadratic residues {{harv|Paley|1933}}.
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| They were introduced as graphs independently by {{harvtxt|Sachs|1962}} and {{harvtxt|Erdős|Rényi|1963}}. [[Horst Sachs|Sachs]] was interested in them for their self-complementarity properties, while [[Paul Erdős|Erdős]] and [[Alfréd Rényi|Rényi]] studied their symmetries.
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| '''Paley digraphs''' are directed analogs of Paley graphs that yield antisymmetric [[conference matrix|conference matrices]]. They were introduced by {{harvtxt|Graham|Spencer|1971}} (independently of Sachs, Erdős, and Rényi) as a way of constructing [[tournament (graph theory)|tournaments]] with a property previously known to be held only by random tournaments: in a Paley digraph, every small [[subset]] of vertices is dominated by some other vertex.
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| == Definition ==
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| Let ''q'' be a [[prime power]] such that ''q'' = 1 (mod 4). That is, ''q'' should either be an arbitrary power of a [[Pythagorean prime]] (a prime congruent to 1 mod 4) or an even power of an odd non-Pythagorean prime. Note that this implies that the unique finite field of order ''q'', '''F'''<sub>''q''</sub>, has a square root of −1.
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| Now let ''V'' = '''F'''<sub>''q''</sub> and
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| :<math>E= \left \{(a,b)\in \mathbf{F}_q\times\mathbf{F}_q \ : \ a-b\in (\mathbf{F}_q^{\times})^2 \right \}</math>.
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| This set is well defined since ''a'' − ''b'' = −(''b'' − ''a''), and since −1 is a square, it follows that ''a'' − ''b'' is a square [[if and only if]] ''b'' − ''a'' is a square.
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| By definition ''G'' = (''V'', ''E'') is the Paley graph of order ''q''.
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| == Example ==
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| For ''q'' = 13, the field '''F'''<sub>''q''</sub> is just integer arithmetic modulo 13. The numbers with square roots mod 13 are:
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| * ±1 (square roots ±1 for +1, ±5 for −1)
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| * ±3 (square roots ±4 for +3, ±6 for −3)
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| * ±4 (square roots ±2 for +4, ±3 for −4).
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| Thus, in the Paley graph, we form a vertex for each of the integers in the range [0,12], and connect each such integer ''x'' to six neighbors: ''x'' ± 1 (mod 13), ''x'' ± 3 (mod 13), and ''x'' ± 4 (mod 13).
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| == Properties ==
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| * The Paley graphs are [[Self-complementary graph|self-complementary]]: the complement of any Paley graph is isomorphic to it, e.g. via the mapping that takes a vertex ''x'' to ''xk'' (mod ''q''), where ''k'' is any nonresidue mod ''q'' {{harv|Sachs|1962}}.
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| * These graphs are [[strongly regular graph]]s with parameters
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| ::<math>srg \left (q, \tfrac{1}{2}(q-1),\tfrac{1}{4}(q-5),\tfrac{1}{4}(q-1) \right ).</math>
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| :In addition, Paley graphs actually form an infinite family of [[conference graph]]s.
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| * The eigenvalues of Paley graphs are <math>\tfrac{1}{2}(q-1)</math> (with multiplicity 1) and <math>\tfrac{1}{2} (-1 \pm \sqrt{q})</math> (both with multiplicity <math>\tfrac{1}{2}(q-1)</math>) and can be calculated using the [[quadratic Gauss sum]].
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| * If q is prime, bounds of the isoperimetric number ''i''(''G'') are:
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| ::<math>\frac{q-\sqrt{q}}{4}\leq i(G) \leq \sqrt { \left (q+\sqrt{q} \right ) \left (\frac{q-\sqrt{q}}{2} \right ) }</math>
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| :This implies that i(G)~O(q), and Paley graph is an [[Expander graph]].
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| * When ''q'' is prime, its Paley graph is a [[Hamiltonian cycle|Hamiltonian]] [[circulant graph]].
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| * Paley graphs are ''quasi-random'' (Chung et al. 1989): the number of times each possible constant-order graph occurs as a subgraph of a Paley graph is (in the limit for large ''q'') the same as for random graphs, and large sets of vertices have approximately the same number of edges as they would in random graphs.
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| == Applications ==
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| * The Paley graph of order 17 is the unique largest graph ''G'' such that neither ''G'' nor its complement contains a complete 4-vertex subgraph (Evans et al. 1981). It follows that the [[Ramsey theory|Ramsey number]] ''R''(4, 4) = 18.
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| * The Paley graph of order 101 is currently the largest known graph ''G'' such that neither ''G'' nor its complement contains a complete 6-vertex subgraph.
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| * Sasukara et al. (1993) use Paley graphs to generalize the construction of the [[Horrocks–Mumford bundle]].
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| ==Paley digraphs==
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| Let ''q'' be a [[prime power]] such that ''q'' = 3 (mod 4). Thus, the finite field of order ''q'', '''F'''<sub>''q''</sub>, has no square root of −1. Consequently, for each pair (''a'',''b'') of distinct elements of '''F'''<sub>''q''</sub>, either ''a'' − ''b'' or ''b'' − ''a'', but not both, is a square. The '''Paley digraph''' is the [[directed graph]] with vertex set ''V'' = '''F'''<sub>''q''</sub> and arc set
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| :<math>A = \left \{(a,b)\in \mathbf{F}_q\times\mathbf{F}_q \ : \ b-a\in (\mathbf{F}_q^{\times})^2 \right \}.</math>
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| The Paley digraph is a [[tournament (graph theory)|tournament]] because each pair of distinct vertices is linked by an arc in one and only one direction.
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| The Paley digraph leads to the construction of some antisymmetric [[conference matrix|conference matrices]] and [[biplane geometries]].
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| == Genus ==
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| The six neighbors of each vertex in the Paley graph of order 13 are connected in a cycle; that is, the graph is [[Neighborhood (graph theory)|locally cyclic]]. Therefore, this graph can be embedded as a [[Triangulation (topology)|Whitney triangulation]] of a [[torus]], in which every face is a triangle and every triangle is a face. More generally, if any Paley graph of order ''q'' could be embedded so that all its faces are triangles, we could calculate the genus of the resulting surface via the [[Euler characteristic]] as <math>\tfrac{1}{24}(q^2 - 13q + 24)</math>. {{harvs|authorlink=Bojan Mohar|last=Mohar|year=2005|txt}} conjectures that the minimum genus of a surface into which a Paley graph can be embedded is near this bound in the case that ''q'' 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
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| :<math>(q^2 - 13q + 24)\left(\tfrac{1}{24} + o(1)\right),</math>
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| where the o(1) term can be any function of ''q'' that goes to zero in the limit as ''q'' goes to infinity.
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| {{harvtxt|White|2001}} finds embeddings of the Paley graphs of order ''q'' ≡ 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.
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| == References ==
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| *{{cite journal
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| | last1 = Baker | first1 = R. D. | last2 = Ebert | first2 = G. L. | last3 = Hemmeter | first3 = J. | last4 = Woldar | first4 = A. J.
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| | title = Maximal cliques in the Paley graph of square order
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| | journal = J. Statist. Plann. Inference
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| | volume = 56
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| | year = 1996
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| | pages = 33–38
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| | doi = 10.1016/S0378-3758(96)00006-7 | ref = harv}}
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| *{{cite journal
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| | doi = 10.1080/16073606.1988.9631945
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| | last1 = Broere | first1 = I. | last2 = Döman | first2 = D. | last3 = Ridley | first3 = J. N.
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| | title = The clique numbers and chromatic numbers of certain Paley graphs
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| | journal = Quaestiones Mathematicae
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| | volume = 11
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| | year = 1988
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| | pages = 91–93 | ref = harv}}
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| *{{cite journal
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| | last1 = Chung | first1 = Fan R. K. | author1-link = Fan Chung
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| | author2-link = Ronald Graham | last2 = Graham | first2 = Ronald L.
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| | last3 = Wilson | first3 = R. M.
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| | title = Quasi-random graphs
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| | journal = [[Combinatorica]]
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| | year = 1989
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| | volume = 9
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| | issue = 4
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| | pages = 345–362
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| | doi = 10.1007/BF02125347 | ref = harv}}
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| *{{Cite journal
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| | last1 = Erdős | first1 = P. | author1-link = Paul Erdős
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| | last2 = Rényi | first2 = A. | author2-link = Alfréd Rényi
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| | doi = 10.1007/BF01895716
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| | mr = 0156334
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| | journal = Acta Mathematica Academiae Scientiarum Hungaricae
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| | pages = 295–315
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| | title = Asymmetric graphs
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| | volume = 14
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| | year = 1963
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| | issue = 3–4 | ref = harv}}
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| *{{cite journal
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| | last1 = Evans | first1 = R. J. | last2 = Pulham | first2 = J. R. | last3 = Sheehan | first3 = J.
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| | title = On the number of complete subgraphs contained in certain graphs
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| | journal = [[Journal of Combinatorial Theory]] | series = Series B
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| | volume = 30
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| | pages = 364–371
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| | year = 1981
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| | doi = 10.1016/0095-8956(81)90054-X
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| | issue = 3 | ref = harv}}
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| *{{Cite journal
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| | last1 = Graham | first1 = R. L. | author1-link = Ronald Graham
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| | last2 = Spencer | first2 = J. H. | author2-link = Joel Spencer
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| | mr = 0292715
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| | journal = [[Canadian Mathematical Bulletin]]
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| | pages = 45–48
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| | title = A constructive solution to a tournament problem
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| | volume = 14
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| | year = 1971
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| | doi = 10.4153/CMB-1971-007-1 | ref = harv}}
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| *{{cite journal
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| | last = Mohar | first = Bojan | authorlink = Bojan Mohar
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| | journal = Electronic Journal of Combinatorics
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| | mr = 2176532
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| | page = N15
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| | title = Triangulations and the Hajós conjecture
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| | url = http://www.combinatorics.org/Volume_12/Abstracts/v12i1n15.html
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| | volume = 12
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| | year = 2005 | ref = harv}}
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| *{{Cite journal
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| | last = Paley | first = R.E.A.C. | authorlink = Raymond Paley
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| | title = On orthogonal matrices
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| | journal = [[J. Math. Phys.]]
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| | volume = 12
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| | pages = 311–320 | ref = harv}}
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| *{{Cite journal
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| | last = Sachs | first = Horst | authorlink = Horst Sachs
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| | mr = 0151953
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| | journal = Publicationes Mathematicae Debrecen
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| | pages = 270–288
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| | title = Über selbstkomplementäre Graphen
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| | volume = 9
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| | year = 1962 | ref = harv}}
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| *{{cite journal
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| | last1 = Sasakura | first1 = Nobuo | last2 = Enta | first2 = Yoichi | last3 = Kagesawa | first3 = Masataka
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| | title = Construction of rank two reflexive sheaves with similar properties to the Horrocks–Mumford bundle
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| | journal = Proc. Japan Acad., Ser. A
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| | volume = 69
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| | issue = 5
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| | pages = 144–148
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| | year = 1993
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| | doi = 10.2183/pjab.69.144 | ref = harv}}
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| *{{cite conference
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| | last = White | first = A. T.
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| | title = Graphs of groups on surfaces
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| | booktitle = Interactions and models
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| | publisher = North-Holland Mathematics Studies 188
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| | location = Amsterdam
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| | year = 2001 | ref = harv}}
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| == External links ==
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| *{{cite web
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| | author= Brouwer, Andries E.
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| | title = Paley graphs
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| | url = http://www.win.tue.nl/~aeb/drg/graphs/Paley.html}}
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| *{{cite web
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| | author = Mohar, Bojan
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| | authorlink = Bojan Mohar
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| | title = Genus of Paley graphs
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| | year = 2005
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| | url = http://www.fmf.uni-lj.si/~mohar/Problems/P0506_PaleyGenus.html}}
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| [[Category:Number theory]]
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| [[Category:Parametric families of graphs]]
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| [[Category:Regular graphs]]
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