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Counting cospectral graphs obtained via switching

Aida Abiad (UGent) , Nils van de Berg and Robin Simoens (UGent)
(2026) DISCRETE MATHEMATICS. 349(3).
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Abstract
Switching is an operation on a graph that does not change the spectrum of the adjacency matrix, thus producing cospectral graphs. An important activity in the field of spectral graph theory is the characterization of graphs by their spectrum. Thus switching provides a tool for disproving the existence of such a characterization. This paper presents a general framework for counting the number of graphs that have a non-isomorphic cospectral graph through a switching method, expanding on the work by Haemers and Spence [European Journal of Combinatorics, 2004]. Our framework is based on a different counting approach, which allows it to be used for all known switching methods for the adjacency matrix. From this, we derive asymptotic results, which we complement with computer enumeration results for graphs up to 10 vertices.
Keywords
graph, eigenvalue, switching, enumeration, REGULAR GRAPHS, GODSIL, CONSTRUCTION, MCKAY

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MLA
Abiad, Aida, et al. “Counting Cospectral Graphs Obtained via Switching.” DISCRETE MATHEMATICS, vol. 349, no. 3, 2026, doi:10.1016/j.disc.2025.114775.
APA
Abiad, A., van de Berg, N., & Simoens, R. (2026). Counting cospectral graphs obtained via switching. DISCRETE MATHEMATICS, 349(3). https://doi.org/10.1016/j.disc.2025.114775
Chicago author-date
Abiad, Aida, Nils van de Berg, and Robin Simoens. 2026. “Counting Cospectral Graphs Obtained via Switching.” DISCRETE MATHEMATICS 349 (3). https://doi.org/10.1016/j.disc.2025.114775.
Chicago author-date (all authors)
Abiad, Aida, Nils van de Berg, and Robin Simoens. 2026. “Counting Cospectral Graphs Obtained via Switching.” DISCRETE MATHEMATICS 349 (3). doi:10.1016/j.disc.2025.114775.
Vancouver
1.
Abiad A, van de Berg N, Simoens R. Counting cospectral graphs obtained via switching. DISCRETE MATHEMATICS. 2026;349(3).
IEEE
[1]
A. Abiad, N. van de Berg, and R. Simoens, “Counting cospectral graphs obtained via switching,” DISCRETE MATHEMATICS, vol. 349, no. 3, 2026.
@article{01K90P5V7PA5C6YE72WHYA2C9R,
  abstract     = {{Switching is an operation on a graph that does not change the spectrum of the adjacency matrix, thus producing cospectral graphs. An important activity in the field of spectral graph theory is the characterization of graphs by their spectrum. Thus switching provides a tool for disproving the existence of such a characterization.
This paper presents a general framework for counting the number of graphs that have a non-isomorphic cospectral graph through a switching method, expanding on the work by Haemers and Spence [European Journal of Combinatorics, 2004]. Our framework is based on a different counting approach, which allows it to be used for all known switching methods for the adjacency matrix. From this, we derive asymptotic results, which we complement with computer enumeration results for graphs up to 10 vertices.}},
  articleno    = {{114775}},
  author       = {{Abiad, Aida and van de Berg, Nils and Simoens, Robin}},
  issn         = {{0012-365X}},
  journal      = {{DISCRETE MATHEMATICS}},
  keywords     = {{graph,eigenvalue,switching,enumeration,REGULAR GRAPHS,GODSIL,CONSTRUCTION,MCKAY}},
  language     = {{eng}},
  number       = {{3}},
  pages        = {{17}},
  title        = {{Counting cospectral graphs obtained via switching}},
  url          = {{http://doi.org/10.1016/j.disc.2025.114775}},
  volume       = {{349}},
  year         = {{2026}},
}
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