On the multiplicities of eigenvalues of graphs and their vertex deleted subgraphs: Old and new results

Abstract

Given a simple graph G, let AG be its adjacency matrix. A principal submatrix of AGof order one less than the order of G is the adjacency matrix of its vertex deleted subgraph. It is well-known that the multiplicity of any eigenvalue of AG and such a principal submatrix can differ by at most one. Therefore, a vertex v of G is a downer vertex (neutral vertex, or Parter vertex) with respect to a fixed eigenvalue μ if the multiplicity of μ in AG−v goes down by one (resp., remains the same, or goes up by one). In this paper, we consider the problems of characterizing these three types of vertices under various constraints imposed on graphs being considered, on vertices being chosen and on eigenvalues being observed. By assigning weights to edges of graphs, we generalize our results to weighted graphs, or equivalently to symmetric matrices.