If the input is PaleyGraph(p) where p is congruent to 1 modulo 4, then the output is an unweighted undirected graph G on p vertices labeled 0,1,...,p-1 where the edge {i,j}, with i<j, is in G iff j-i is a quadratic residue in Zp.

If the input is PaleyGraph(p) where p is congruent to 3 modulo 4, then the output is an unweighted directed graph G on p vertices labeled 0,1,...,p-1 where the arc [i,j] is in G iff j-i is a quadratic residue in Zp.

If the input is PaleyGraph(p,k) where $q=p^k$ is congruent to 1 modulo 4, then the output is an unweighted undirected graph G on $q$ vertices labeled 0,1,...,q-1 where the edge {i,j}, i<j, is in G iff y-x is a square in the finite field GF(q), where x is the $i$th element and y is the $j$th element in GF(q). The numbering of the elements in GF(q) is lexicographical.

Similarly, if the input is PaleyGraph(p,k) where $q=p^k$ is congruent to 3 modulo 4, then the output is an unweighted directed graph G on $q$ vertices labeled 0,1,...,q-1 where the arc [i,j] is in G iff y-x is a square in the finite field GF(q), where x is the $i$th element and y is the $j$th element in GF(q). The numbering of the elements in GF(q) is lexicographical.

The vertex label for the element f(x) in Zp[x] is f(p). For example, in GF(23) the elements are ordered $0,1,x,x+1,x^2,x^2+1,x^2+x,x^2+x+1$. Thus the label for element $x^2+1$ is $2^2+1=5$.

The field can be specified by the user by specifying the extension polynomial for GF(q), a monic irreducible polynomial m(x) in Zp[x] of degree k.

$\mathrm{with}\left(\mathrm{GraphTheory}\right)\&colon;$

$\mathrm{with}\left(\mathrm{SpecialGraphs}\right)\&colon;$

$P≔\mathrm{PaleyGraph}\left(5\right)$

$P≔\mathrm{Graph\; 1:\; an\; undirected\; graph\; with\; 5\; vertices\; and\; 5\; edge(s)}$

$\mathrm{DrawGraph}\left(P\right)$

$E≔\mathrm{Edges}\left(P\right)$

$E≔\left\{\left\{0\&comma;1\right\}\&comma;\left\{0\&comma;4\right\}\&comma;\left\{1\&comma;2\right\}\&comma;\left\{2\&comma;3\right\}\&comma;\left\{3\&comma;4\right\}\right\}$

$P≔\mathrm{PaleyGraph}\left(2\&comma;2\right)$

$P≔\mathrm{Graph\; 2:\; an\; undirected\; graph\; with\; 4\; vertices\; and\; 6\; edge(s)}$

$\mathrm{DrawGraph}\left(P\right)$

$E≔\mathrm{Edges}\left(P\right)$

$E≔\left\{\left\{0\&comma;1\right\}\&comma;\left\{0\&comma;2\right\}\&comma;\left\{0\&comma;3\right\}\&comma;\left\{1\&comma;2\right\}\&comma;\left\{1\&comma;3\right\}\&comma;\left\{2\&comma;3\right\}\right\}$

$P≔\mathrm{PaleyGraph}\left(3\&comma;2\&comma;y^2+1\right)$

$P≔\mathrm{Graph\; 3:\; an\; undirected\; graph\; with\; 9\; vertices\; and\; 18\; edge(s)}$

$\mathrm{DrawGraph}\left(P\right)$