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8 - Polynomial Reconstruction

Published online by Cambridge University Press:  15 May 2026

Zoran Stanić
Affiliation:
University of Belgrade, Serbia
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Summary

This chapter examines the spectral version of the well-known Ulam reconstruction conjecture, focusing on whether the characteristic polynomial of the adjacency matrix of a signed graph with at least three vertices is uniquely determined by the collection of characteristic polynomials of its vertex-deleted subgraphs. In the setting of signed graphs, a simple counterexample already exists: a pair of signed cycles on the same vertex set, with exactly one of them balanced. Beyond this instance, no additional counterexamples are known. We establish a sequence of structural properties that are reconstructible from the polynomial deck and identify several classes of signed graphs for which the characteristic polynomial is uniquely reconstructible. These classes include trees, unicyclic signed graphs (other than cycles) and signed graphs whose spectrum is bounded below by −2, among others. We also present numerous specific results concerning additional graph families.

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  • Polynomial Reconstruction
  • Zoran Stanić, University of Belgrade, Serbia
  • Book: Spectra of Signed Graphs
  • Online publication: 15 May 2026
  • Chapter DOI: https://doi.org/10.1017/9781009853316.009
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  • Polynomial Reconstruction
  • Zoran Stanić, University of Belgrade, Serbia
  • Book: Spectra of Signed Graphs
  • Online publication: 15 May 2026
  • Chapter DOI: https://doi.org/10.1017/9781009853316.009
Available formats
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Save book to Google Drive

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Google Drive.

  • Polynomial Reconstruction
  • Zoran Stanić, University of Belgrade, Serbia
  • Book: Spectra of Signed Graphs
  • Online publication: 15 May 2026
  • Chapter DOI: https://doi.org/10.1017/9781009853316.009
Available formats
×