The 
-Peisert graph, called a 
graph by Peisert (2001), is a
conference graph defined by taking a vertex
set consisting of elements of the finite field
GF(
), with 
and 
even, and an edge set consisting
of, for some fixed primitive root 
of GF(
), all elements 
in GF(
) that satisfy 
for all possible 
. This construction gives the same graph
regardless of the choice of primitive root 
(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 all other orders
are distinct graphs not isomorphic to Paley
graphs but cospectral with them (Alexander
2015). In addition, for the unique sporadic case of 529 vertices,
there is an additional graph that is cospectral
with both the 529-Paley graph and 529-Peisert graphs
but is isomorphic to neither (Peisert 2001).
Peisert graphs are self-complementary, vertex-transitive, edge-transitive, arc-transitive, distance-regular, and distance-transitive.
Peisert graphs (and the sporadic 529-vertex Peisert graph) are their own graph distance-2 graphs.
Peisert graphs are implemented in the Wolfram Language as GraphData[
"Peisert", q
] for small orders 
.
