Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: 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.

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The categories of this paper are: acc-phys
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The categories of this paper are: q-fin
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The categories of this paper are: cs
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The categories of this paper are: bayes-an
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The categories of this paper are: chao-dyn
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The categories of this paper are: ao-sci
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The categories of this paper are: q-bio
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The categories of this paper are: hep-ex
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The categories of this paper are: econ
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The categories of this paper are: eess
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The categories of this paper are: stat
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The categories of this paper are: patt-sol
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The categories of this paper are: nucl-th
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The categories of this paper are: astro-ph
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The categories of this paper are: q-alg
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The categories of this paper are: gr-qc