The 
-Peisert graph, called a 
graph by Peisert (2001), is a
conference graph with vertex
set the finite field GF(
), where 
is a prime with 
and 
is an even positive
integer. Choose a generator 
of the multiplicative
group of nonzero elements
of GF(
).
The edge set consists of the unordered pairs 
for which 
for some integer 
congruent
to 0 or 1 modulo 4. Different choices of 
give isomorphic graphs
(Peisert 2001, Alexander 2015).
As a result of the restriction on 
, Peisert graphs are defined for 
, 49, 81, 121, 361, 529, 729, 961, ... (OEIS A383487).
The 9-Peisert graph is isomorphic to the 9-Paley graph (and the generalized
quadrangle 
),
but for every other allowed graph order, the Peisert
graph is cospectral with the Paley
graph of the same graph order and is not isomorphic
to it (Alexander 2015). In the classification of self-complementary symmetric graphs, there is one additional sporadic
graph, with 529 vertices.
It is cospectral with both the 529-Paley
graph and the 529-Peisert graph and is isomorphic
to neither (Peisert 2001).
Peisert graphs are self-complementary, vertex-transitive, edge-transitive, arc-transitive, distance-regular, and distance-transitive.
The distance-2 graph distance graph of a Peisert graph (including the sporadic 529-vertex graph) is its graph complement, which is isomorphic to the original graph.
The strongly regular parameters are 
, and the
least graph eigenvalue is 
. The Hoffman
ratio bound therefore gives a chromatic number
of at least 
.
For 
,
where 
is an odd prime power satisfying 
, this bound is attained and
 |
where 
,
, and 
denote the clique number,
independence number, and chromatic
number, respectively. In this case, the Peisert graph is obtained from the affine plane over GF(
) by selecting 
directions (Brouwer et al. 2026). The lines in
a selected direction form cliques of size 
, while the lines in an omitted direction partition the vertices into 
independent vertex sets.
These provide a matching lower bound and upper
bound on the chromatic number.
The sporadic 529-vertex Peisert graph has the same graph spectrum as the 529-Peisert graph, so its chromatic number is at least 23 by the same Hoffman ratio bound.
For all sufficiently large admissible 
, Timmons (2026) proved that the achromatic
number of the 
-Peisert
graph is 
.
Unlike the chromatic number, this maximizes the
number of colors in a proper vertex coloring in
which every pair of colors occurs at the ends of some graph
edge. Timmons (2026) reports ChatGPT assistance with mathematical proofs.
Peisert graphs are implemented in the Wolfram Language as GraphData[
"Peisert", q
] for small graph
orders 
.
