The minimum number of eigenvalues of multiplicity one in a diagonalizable matrix, over a field, whose graph is a tree

It is known that an n-by-n Hermitian matrix, n≥2, whose graph is a tree necessarily has at least two eigenvalues (the largest and smallest, in particular) with multiplicity 1. Recently, much of the multiplicity theory, for eigenvalues of Hermitian matrices whose graph is a tree, has been generalized to geometric multiplicities of eigenvalues of matrices over a general field (whose graph is a tree). However, the two 1's fact does not generalize. Here, we give circumstances under which there are two 1's and give several examples (without two 1's) that limit our positive results.