The minimum number of eigenvalues of multiplicity one in a diagonalizable matrix, over a field, whose graph is a tree
The first and the third author [Linear Algebra Appl. 537, 332--347 (2018; Zbl 1376.15030)] recently began to generalize the multiplicity theory for Hermitian matrices and trees to geometric multiplicities of general matrices over a field when the graph is a tree. In the present paper, the authors are interested in the number of eigenvalues of multiplicity 1 in the case of diagonalizable matrices over a general field when the graph is a tree. They give several positive results that show that at least two multiplicity 1 eigenvalues exist if the tree has either a small number of vertices of degree \(\geq 3\) or a small number of \(g\)-Parter vertices or a small diameter, accompanied by examples featuring at most one multiplicity 1 eigenvalue in the opposite cases.
scientific article; zbMATH DE number 6954362