Paley graph

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. This choice of q implies that in the unique finite field F_q of order q, the element  −1 has a square root.

Now let V = F_q and let

${\displaystyle E=\left\{\{a,b\}\$ :\ a-b\in (\mathbf {F} _{q}^{\times })^{2}\right\}} .

If a pair {a,b} is included in E, it is included under either ordering of its two elements. For, a − b = −(b − a), and  −1 is a square, from which it follows that a − b is a square if and only if b − a is a square.

By definition G = (V, E) is the Paley graph of order q.

For q = 13, the field F_q is just integer arithmetic modulo 13. The numbers with square roots mod 13 are:

• ±1 (square roots ±1 for +1, ±5 for −1)
• ±3 (square roots ±4 for +3, ±6 for −3)
• ±4 (square roots ±2 for +4, ±3 for −4).

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).

The Paley graphs are self-complementary: the complement of any Paley graph is isomorphic to it. One isomorphism is via the mapping that takes a vertex x to xk (mod q), where k is any nonresidue mod q (Sachs 1962).

Paley graphs are strongly regular graphs, with parameters

${\displaystyle srg\left(q,{\tfrac {1}{2}}(q-1),{\tfrac {1}{4}}(q-5),{\tfrac {1}{4}}(q-1)\right).}$

This in fact follows from the fact that the graph is arc-transitive and self-complementary. In addition, Paley graphs form an infinite family of conference graphs.

The eigenvalues of Paley graphs are ${\displaystyle {\tfrac {1}{2}}(q-1)}$ (with multiplicity 1) and ${\displaystyle {\tfrac {1}{2}}(-1\pm {\sqrt {q}})}$ (both with multiplicity ${\displaystyle {\tfrac {1}{2}}(q-1)}$). They can be calculated using the quadratic Gauss sum or by using the theory of strongly regular graphs.

If q is prime, the isoperimetric number i(G) of the Paley graph is known to satisfy the following bounds:

${\displaystyle {\frac {q-{\sqrt {q}}}{4}}\leq i(G)\leq {\sqrt {\left(q+{\sqrt {q}}\right)\left({\frac {q-{\sqrt {q}}}{2}}\right)}}.}$

When q is prime, the associated Paley graph is a Hamiltonian circulant graph.

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.

• The Paley graph of order 9 is a locally linear graph, a rook's graph, and the graph of the 3-3 duoprism.
• The Paley graph of order 13 has book thickness 4 and queue number 3 (Wolz 2018).
• 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 number R(4, 4) = 18.
• 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.
• Sasukara et al. (1993) use Paley graphs to generalize the construction of the Horrocks–Mumford bundle.

Let q be a prime power such that q = 3 (mod 4). Thus, the finite field of order q, F_q, has no square root of −1. Consequently, for each pair (a,b) of distinct elements of F_q, either a − b or b − a, but not both, is a square. The Paley digraph is the directed graph with vertex set V = F_q and arc set

${\displaystyle A=\left\{(a,b)\in \mathbf {F} _{q}\times \mathbf {F} _{q}\$ :\ b-a\in (\mathbf {F} _{q}^{\times })^{2}\right\}.}

The Paley digraph is a tournament because each pair of distinct vertices is linked by an arc in one and only one direction.

The Paley digraph leads to the construction of some antisymmetric conference matrices and biplane geometries.

The six neighbors of each vertex in the Paley graph of order 13 are connected in a cycle; that is, the graph is locally cyclic. Therefore, this graph can be embedded as a 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 ${\displaystyle {\tfrac {1}{24}}(q^{2}-13q+24)}$. Mohar (2005) 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

${\displaystyle (q^{2}-13q+24)\left({\tfrac {1}{24}}+o(1)\right),}$

where the o(1) term can be any function of q that goes to zero in the limit as q goes to infinity.

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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