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Inverse eigenvalue and related problems for hollow matrices described by graphs

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F. Scott Dahlgren, Zachary Gershkoff, Leslie Hogben, Sara Motlaghian, Derek Young

Combinatorics · Submitted 5 Feb 2022

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim Solutions to the hollow inverse eigenvalue problems for paths and complete bipartite graphs are presented.

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arXiv ID2202.02633 (version 2)
Pith ID3ESKILCY, Pith's permanent record of this paper and its review
CitationDerek Young, F. Scott Dahlgren, Leslie Hogben, Sara Motlaghian, Zachary Gershkoff. "Inverse eigenvalue and related problems for hollow matrices described by graphs." arXiv:2202.02633 (2022). https://pith.science/paper/3ESKILCY
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What the abstract itself states, as probabilities: claims first 0.03 · states limitation 0.08 · human participants 0.01 · controlled comparison 0.01 · randomized assignment 0.01 · states quantitative result 0.09 · compares with prior results 0.06 · mentions code or data availability 0.02.

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arXiv listing abstract v2. Results and the main claim are underlined; .

A hollow matrix described by a graph $G$ is a real symmetric matrix having all diagonal entries equal to zero and with the off-diagonal entries governed by the adjacencies in $G$. For a given graph $G$, the determination of all possible spectra of matrices associated with $G$ is the hollow inverse eigenvalue problem for $G$. Solutions to the hollow inverse eigenvalue problems for paths and complete bipartite graphs are presented. Results for related subproblems such as possible ordered multiplicity lists, maximum multiplicity of an eigenvalue, and minimum number of distinct eigenvalues are presented for additional families of graphs.

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#sentencerolethe authors' own workstates the main conclusionstrength
s0 A hollow matrix described by a graph $G$ is a real symmetric matrix having all diagonal entries equal to zero and with the off-diagonal entries governed by the adjacencies in $G$. background 1.00 0.06 0.03 asserted 1.00
s1 For a given graph $G$, the determination of all possible spectra of matrices associated with $G$ is the hollow inverse eigenvalue problem for $G$. background 1.00 0.08 0.08 asserted 1.00
s2 Solutions to the hollow inverse eigenvalue problems for paths and complete bipartite graphs are presented. contribution 0.55 0.90 0.48 asserted 1.00
s3 Results for related subproblems such as possible ordered multiplicity lists, maximum multiplicity of an eigenvalue, and minimum number of distinct eigenvalues are presented for additional families of graphs. result 0.65 0.88 0.14 asserted 1.00

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 1.00.

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Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

The paper
Original abstract

A hollow matrix described by a graph $G$ is a real symmetric matrix having all diagonal entries equal to zero and with the off-diagonal entries governed by the adjacencies in $G$. For a given graph $G$, the determination of all possible spectra of matrices associated with $G$ is the hollow inverse eigenvalue problem for $G$. Solutions to the hollow inverse eigenvalue problems for paths and complete bipartite graphs are presented. Results for related subproblems such as possible ordered multiplicity lists, maximum multiplicity of an eigenvalue, and minimum number of distinct eigenvalues are presented for additional families of graphs.

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