The minimum number of distinct eigenvalues 
of an 
-vertex simple graph 
is the least number of distinct eigenvalues
of a real 
symmetric matrix 
whose off-diagonal entry 
is nonzero iff 
is an graph edge of 
(Ahmadi et al. 2013). The diagonal
entries are unrestricted. The adjacency matrix
is one admissible choice, so 
is at most the number of distinct values in the graph
spectrum.
An empty graph of positive order has 
, as witnessed by the identity
matrix. For the complete graph 
with 
,
the matrix with every entry equal to 1 has exactly two
distinct eigenvalues. A real symmetric
matrix with only one distinct eigenvalue is a
scalar multiple of the identity matrix, so 
.
Nguyen (2026) proved the sparse-complement conjecture. If 
has 
vertices and its graph complement has at most 
edges, then 
.
