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 $\mathbf{m}=(m_1,\dots ,m_{\ell })$ 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 IEPG. Recent work solved the ordered multiplicity IEPG for all graphs on 6 vertices.

In this work, we develop zero forcing methods for the ordered multiplicity IEPG 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 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 IEPG 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.

给定一个图G，人们可能会问：“ G 的所有加权邻接矩阵可能有哪些特征值集？” （边的权重为正或负，而对角线项可以是任何实数。）这被称为图的逆特征值问题 (IEPG)。这个问题的温和放松考虑了多重性列表而不是确切的特征值本身。也就是说，给定一个在n个顶点上的图G和一个有序的分区$\mathbf{米}=({米}_1,\dots ,{米}_{\ell })$的n，是否存在加权邻接矩阵，其中第i个不同的特征值具有多重性${米}_{\mathrm{一世}}$? 这称为有序多重性 IEPG。最近的工作解决了 6 个顶点上所有图的有序多重性 IEPG。

在这项工作中，我们为多种不同环境中的有序多重性 IEPG 开发了迫零方法。也就是说，我们利用图的幂的零强迫参数来实现连续多重性的界限。我们能够提供图的特征值的多重性总和的一般界限。这包括对连续特征值的多重性总和的新界限以及更具体的树界限。使用这些结果，我们验证了上面关于六个顶点上的 IEPG 的先前结果。此外，将我们的技术应用于斜对称矩阵，我们能够为五个顶点上的连通图确定斜对称矩阵的所有可能的有序重数列表。