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 = Summary: The minimum number of distinct eigenvalues of a graph Notation: · 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 an edge. · 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 [5] · There exist trees with q(T) > d(T) + 1; see [1] · 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 [3] · More work on general graphs; see [2] for example. Questions: (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? Collections: Mathematics