Paley graph

Paley graph
The Paley graph of order 13
Named afterRaymond Paley
Verticesq ≡ 1 mod 4,
q prime power
Edgesq(q − 1)/4
Diameter2
PropertiesStrongly regular
Conference graph
Self-complementary
NotationQR(q)
Table of graphs and parameters

In mathematics, Paley graphs are undirected graphs constructed from the members of a suitable finite field by connecting pairs of elements that differ by a quadratic residue. The Paley graphs form an infinite family of conference graphs, which yield an infinite family of symmetric 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.

Paley graphs are named after Raymond Paley. They are closely related to the Paley construction for constructing Hadamard matrices from quadratic residues.[1] They were introduced as graphs independently by Sachs (1962) and Erdős & Rényi (1963). Sachs was interested in them for their self-complementarity properties,[2] while Erdős and Rényi studied their symmetries.[3]

Paley digraphs are directed analogs of Paley graphs that yield antisymmetric conference matrices. They were introduced by Graham & Spencer (1971) (independently of Sachs, Erdős, and Rényi) as a way of constructing 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.[4]

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  4. ^ Cite error: The named reference gs was invoked but never defined (see the help page).