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PreprintCC BY 4.0arXiv

The switching conjecture for main eigenvalues is asymptotically true

  • Saieed Akbari
  • Hitesh Kumar
  • Bojan Mohar
  • Shivaramakrishna Pragada
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Abstract

An eigenvalue of a signed graph is called \emph{main} if there exists a corresponding eigenvector non-orthogonal to the all-ones vector. An important result of O'Rourke and Touri (2016) states that almost all (unsigned) graphs have all main eigenvalues. Akbari, França, Ghasemian, Javarsineh, and de Lima (2021) considered main eigenvalues of signed graphs and conjectured that for any unsigned connected graph G ∉{ K2, K4 − e}, there is a switching s such that all eigenvalues of the signed graph Gs are main. We prove two incomparable asymptotic versions of this conjecture. We show that for any graph G of order n, there exists a switching s∈{±1}n such that Gs has n − O (n/((log n)1/4)) main eigenvalues counted with multiplicity. Using a similar proof strategy, we also show that if G has d distinct eigenvalues, then there exists a switching s∈{±1}n such that Gs has d − O (d/((log d)1/4)) main eigenvalues.

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Cite

BibTeXRIS
BibTeX
@misc{akbari2026switching,
  title = {{The switching conjecture for main eigenvalues is asymptotically true}},
  author = {Saieed Akbari and Hitesh Kumar and Bojan Mohar and Shivaramakrishna Pragada},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.27046},
  eprint = {2609.27046},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.27046},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - The switching conjecture for main eigenvalues is asymptotically true
AU  - Saieed Akbari
AU  - Hitesh Kumar
AU  - Bojan Mohar
AU  - Shivaramakrishna Pragada
PY  - 2026
DA  - 2026/09/22
PB  - arXiv
DO  - 10.48550/arxiv.2609.27046
UR  - https://arxiv.org/abs/2609.27046
AB  - An eigenvalue of a signed graph is called \emph{main} if there exists a corresponding eigenvector non-orthogonal to the all-ones vector. An important result of O'Rourke and Touri (2016) states that almost all (unsigned) graphs have all main eigenvalues. Akbari, França, Ghasemian, Javarsineh, and de Lima (2021) considered main eigenvalues of signed graphs and conjectured that for any unsigned connected graph G ∉{ K_2, K_4 − e}, there is a switching s such that all eigenvalues of the signed graph G^s are main. We prove two incomparable asymptotic versions of this conjecture. We show that for any graph G of order n, there exists a switching s∈{±1}^n such that G^s has n − O (n/((log n)^1/4)) main eigenvalues counted with multiplicity. Using a similar proof strategy, we also show that if G has d distinct eigenvalues, then there exists a switching s∈{±1}^n such that G^s has d − O (d/((log d)^1/4)) main eigenvalues.
ER  -