Constructing Laplacian matrices with Soules vectors: inverse eigenvalue problem and applications

The symmetric nonnegative inverse eigenvalue problem (SNIEP) asks which sets of numbers (counting multiplicities) can be the eigenvalues of a symmetric matrix with nonnegative entries. While examples of such matrices are abundant in linear algebra and various applications, this question is still open for matrices of dimension $N\geq 5$. One of the approaches to solve the SNIEP was proposed by George W. Soules, relying on a specific type of eigenvectors (Soules vectors) to derive sufficient conditions for this problem. Elsner et al. later showed a canonical way to construct all Soules vectors, based on binary rooted trees. While Soules vectors are typically treated as a totally ordered set of vectors, we propose in this article to consider a relaxed alternative: a partially ordered set of Soules vectors. We show that this perspective enables a more complete characterization of the sufficient conditions for the SNIEP. In particular, we show that the set of eigenvalues that satisfy these sufficient conditions is a convex cone, with symmetries corresponding to the automorphisms of the binary rooted tree from which the Soules vectors were constructed. As a second application, we show how Soules vectors can be used to construct graph Laplacian matrices with a given spectrum and describe a number of interesting connections with the concepts of hierarchical random graphs, equitable partitions and effective resistance.