  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.
