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University of Illinois Urbana-Champaign
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Some results on symmetric signings
Carlson, Charles A
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https://hdl.handle.net/2142/98401
Description
Title
Some results on symmetric signings
Author(s)
Carlson, Charles A
Issue Date
2017-07-17
Director of Research (if dissertation) or Advisor (if thesis)
Kolla, Alexandra
Department of Study
Computer Science
Discipline
Computer Science
Degree Granting Institution
University of Illinois at Urbana-Champaign
Degree Name
M.S.
Degree Level
Thesis
Date of Ingest
2017-09-29T17:56:57Z
Keyword(s)
Matrix signings
Spectral graph theory
Eigenvalues
Matchings
Determinant
Abstract
In this work, we investigate several natural computational problems related to identifying symmetric signings of symmetric matrices with specific spectral properties. We show NP-completeness for verifying whether an arbitrary matrix has a symmetric signing that is positive semi-definite, is singular, or has bounded eigenvalues. We exhibit a stark contrast between invertibility and the above-mentioned spectral properties by presenting a combinatorial characterization of matrices with invertible symmetric signings and an efficient algorithm using this characterization to verify whether a given matrix has an invertible symmetric signing. Finally, we give efficient algorithms to verify and find invertible and singular symmetric signing for matrices whose support graph is bipartite.
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