Summary: The minimum number of distinct eigenvalues of a graph
· For a graph G on n vertices, S(G) denotes the set of real symmetric matrices A =
[aij] Mn such that aij = 0, (i = j) if and only if vertices i and j are connected by
· d(G) denotes the diameter of the graph G.
· q(G): The minimum number of distinct eigenvalues of G when minimum is taken over
all matrices in S(G) .
Some of the known results:
· For any tree T, q(T) d(T) + 1; see 
· There exist trees with q(T) > d(T) + 1; see 
· For any positive integer d, there exists a constant f(d) such that for any tree T with
diameter d, there is a matrix A S(T) with at most f(d) distinct eigenvalues (claimed
by B. Shader who also says:"our f(d) is super super exponential").
· f(d) (9/8)d for d large; see 
· More work on general graphs; see  for example.
(1) What about graphs with diameter d? Is there any constant similar to f(d)?
(2) Is f(d) linear in d or exponential?
(3) What is q(T) if T is a complete binary tree of height h?