The minimum number of distinct eigenvalues among the symmetric matrices with a given graph

A classic result is that the number of distinct eigenvalues of a graph of diameter d is at least d+1. More generally, let G be a graph with n vertices, and let S(G) denote the set of all symmetric n by n matrices, A = [aij ], with the property that for off-diagonal entries, aij is nonzero if and only if G has an edge from i to j. The traditional proof of the classic result extends to A ∈ S(G). Thus, if A ∈ S(G), then A has at least d + 1 distinct eigenvalues. Saiago and Johnson recently conjectured that if T is a tree of diameter d, then there exists a matrix in S(T ) with exactly d+ 1 distinct eigenvalues. More recently, Barioli and Fallat have disproved the conjecture with a tree of diameter 7, such that every matrix in S(T ) has at least 9 distinct eigenvalues. We use the Smith Normal Form for matrices with polynomial entries, to give an inifinite family of trees T of diameter d such that every A ∈ S(T ) has at least (9/8)d distinct eigenvalues. We also show that for each d, there is an integer f(d), such that every tree T of diameter d (but an arbitrary number of vertices) there exists a matrix A ∈ S(T ) with at most f(d) distinct eigenvalues. Finally, we discuss bounds on f(d) for small d.