A zero forcing technique for bounding sums of eigenvalue multiplicities

Abstract: 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 (IEP-$G$). 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 $\mathbf{m}= (m_1, \ldots, m_\ell)$ of $n$, is there a weighted adjacency matrix where the $i$-th distinct eigenvalue has multiplicity $m_i$? This is known as the ordered multiplicity IEP-$G$. Recent work solved the ordered multiplicity IEP-$G$ for all graphs on 6 vertices. In this work, we develop zero forcing methods for the ordered multiplicity IEP-$G$ in a multitude of different contexts. Namely, we utilize zero forcing parameters on powers of graphs to achieve bounds on consecutive multiplicities. We are able to provide general bounds on sums of multiplicities of eigenvalues for graphs. This includes new bounds on the the sums of multiplicities of consecutive eigenvalues as well as more specific bounds for trees. Using these results, we verify the previous results above regarding the IEP-$G$ on six vertices. In addition, applying our techniques to skew-symmetric matrices, we are able to determine all possible ordered multiplicity lists for skew-symmetric matrices for connected graphs on five vertices.