The Brouwer-Haemers graph is the unique strongly regular graph on 81 vertices with parameters 
, 
, 
, 
(Brouwer and Haemers 1992, Brouwer). It is also distance-regular
with intersection array 
, as well as distance-transitive.
This graph can be constructed with vertices corresponding to polynomials in the finite field GF(81) where two points are adjacent when their difference is a fourth power (Brouwer), making it a quartic analog of the cyclotomic graphs and Paley graphs.
It is also the local graph of the generalized quadrangle 
,
i.e., the vertex-induced subgraph on 
by the neighbors of any single
vertex.
It has graph spectrum 
and is therefore an integral
graph. It has graph automorphism group
order 
and chromatic number 7.
The Brouwer-Haemers graph is implemented in the Wolfram Language as GraphData["BrouwerHaemersGraph"].
