If the input is PayleyGraph(p) then the output is an undirected unweighted simple 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 PayleyGraph(p,k) then the output is an undirected unweighted simple graph G on  vertices labeled 0,1,...,q-1 where the edge {i,j}, i<j, is in G iff y-x is a square the finite field GF(q) where x is the th element and y is the 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 . Thus the label for element  is .

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