Diminimal Families of Arbitrary Diameter

Given a tree T, let q(T) be the minimum number of distinct eigenvalues in a symmetric matrix whose underlying graph is T. It is well known that q(T)≥d(T)+1, where d(T) is the diameter of T, and a tree T is said to be diminimal if q(T)=d(T)+1. In this paper, we present families of diminimal trees of any fixed diameter. Our proof is constructive, allowing us to compute, for any diminimal tree T of diameter d in these families, a symmetric matrix M with underlying graph T whose spectrum has exactly d+1 distinct eigenvalues.