The Kesten-McKay distribution is the limiting empirical adjacency-eigenvalue distribution of uniformly random regular graphs
with fixed degree 
. In the unscaled adjacency convention its density is
 |
for 
and is zero outside this interval. It is also the spectral
measure at a vertex of
the infinite 
-regular tree. For a finite 
-regular graph on 
vertices, the constant eigenvector has eigenvalue 
and contributes mass 
to the empirical measure.
