Classification of vertices and edges with respect to the geometric multiplicity of an eigenvalue in a matrix, with a given graph, over a field

Abstract

We are interested in the geometric multiplicity of an identified eigenvalue of a matrix A over a general field with a given graph G for its off-diagonal entries. By the classification of a vertex (edge) of G, we refer to the change in the geometric multiplicity of when this vertex (edge) is removed from G to leave a principal submatrix (modification) of A. Such classification in the case of Hermitian matrices and trees has been strategic in the problem of determining the possible lists of multiplicities for the eigenvalues among matrices with the given graph. Here, our view of, and qualification of, the classification in the general setting provides a tool that makes some past arguments more transparent and provides new insight in both the classical and general setting. Some applications are given to general downer mechanisms for the recognition of Parter vertices and to the stability of geometric multiplicity under perturbation of a diagonal entry.