Let be a prime power such that . Note that this implies that the unique finite field of order , , has a square root of . Now let and

By hypothesis, if and only if . By definition is the Paley graph of order .

Paley was a brilliant mathematician who died tragically at the age of 26. Paley graphs are one of the many spin-offs of his work. The following facts are known about them.

1. The eigenvalues of Paley graphs are (with multiplicity ) and
   (both with multiplicity ).
2. It is known that a Paley graph is a Ramanujan graph.
3. It is known that the family of Paley graphs of prime order is a vertex expander graph family.
4. If , where is prime, then has order .

Here is Sage code for the Paley graph (thanks to Chris Godsil, see [GB]):

def Paley(q):
    K = GF(q)
    return Graph([K, lambda i,j: i != j and (i-j).is_square()])

(Replace “K” by “” above; I was having trouble rendering it in html.) Below is an example.

sage: X = Paley(13)
sage: X.vertices()
[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12]
sage: X.is_vertex_transitive()
True
sage: X.degree_sequence()
[6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6]
sage: X.spectrum()
[6, 1.302775637731995?, 1.302775637731995?, 1.302775637731995?,
1.302775637731995?, 1.302775637731995?, 1.302775637731995?,
-2.302775637731995?, -2.302775637731995?, -2.302775637731995?,
-2.302775637731995?, -2.302775637731995?, -2.302775637731995?]
sage: G = X.automorphism_group()
sage: G.cardinality()
78

We see that this Paley graph is regular of degree , it has only three distinct eigenvalues, and its automorphism group is order .

Here is an animation of this Paley graph:

The frames in this animation were constructed one-at-a-time by deleting an edge and plotting the new graph.

Here is an animation of the Paley graph of order :

The frames in this animation were constructed using a Python script:

X = Paley(17)
E = X.edges()
N = len(E)
EC = X.eulerian_circuit()
for i in range(N):
    X.plot(layout="circular", graph_border=True, dpi=150).save(filename="paley-graph_"+str(int("1000")+int("%s"%i))+".png")
    X.delete_edge(EC[i])
X.plot(layout="circular", graph_border=True, dpi=150).save(filename="paley-graph_"+str(int("1000")+int("%s"%N))+".png")

Instead of removing the frames “by hand” they are removed according to their occurrence in a Eulerian circuit of the graph.

Here is an animation of the Paley graph of order :

[GB] Chris Godsil and Rob Beezer, Explorations in Algebraic Graph Theory with Sage, 2012, in preparation.