A zero forcing technique for bounding sums of eigenvalue multiplicities

Given a graph G, one may ask: “What sets of eigenvalues are possible over all weighted adjacency matrices of G?” (The weight of an edge is positive or negative, while the diagonal entries can be any real numbers.) This is known as the Inverse Eigenvalue Problem for graphs (IEPG). A mild relaxation of this question considers the multiplicity list instead of the exact eigenvalues themselves. That is, given a graph G on n vertices and an ordered partition m=(m1,…,mℓ) of n, is there a weighted adjacency matrix where the i-th distinct eigenvalue has multiplicity mi? This is known as the ordered multiplicity IEPG. Recent work solved the ordered multiplicity IEPG for all graphs on 6 vertices.