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 and 529-Pesiert graphs but is isomorphic to neither (Peisert 2001).
Peisert graphs are self-complementary, vertex-transitive, egde-transitive, arc-transitive, distance-regular, and distance-transitive.
Peisert graphs (and the sporadic 529-vertex Peisert graph) sare their own graph distance-2 graphs.
Peisert graphs will be implemented in a future version of the Wolfram Language as GraphData[
"Peisert", q
] for small orders 
.
