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PreprintCC0arXiv

The least signless Laplacian eigenvalue of {C3,C5}-free graphs

  • Qi Zhou
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Abstract

Brandt [Discrete Math. 183 (1998) 17--25] conjectured that the least signless Laplacian eigenvalue of every regular triangle-free graph of order n is at most 4n/25. Using flag algebras, Balogh, Clemen, Lidický, Norin and Volec [SIAM J. Discrete Math. 37 (2023) 1173--1179] established the stronger bound 15n/94 for all triangle-free graphs. We investigate the effect of additionally excluding pentagons and prove that every {C3,C5}-free graph G of order n satisfies qmin(G)<0.0569n, without any regularity assumption. The proof combines seven-vertex flag inequalities with local Rayleigh constraints that retain the least eigenvalue throughout the counting argument. An exact integer certificate establishes the required inequality.

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Cite

BibTeXRIS
BibTeX
@misc{zhou2026least,
  title = {{The least signless Laplacian eigenvalue of $\C_3,C_5\$-free graphs}},
  author = {Qi Zhou},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.00875},
  eprint = {2610.00875},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.00875},
  note = {CC0},
}
RIS
TY  - UNPB
TI  - The least signless Laplacian eigenvalue of {C_3,C_5}-free graphs
AU  - Qi Zhou
PY  - 2026
DA  - 2026/10/01
PB  - arXiv
DO  - 10.48550/arxiv.2610.00875
UR  - https://arxiv.org/abs/2610.00875
AB  - Brandt [Discrete Math. 183 (1998) 17--25] conjectured that the least signless Laplacian eigenvalue of every regular triangle-free graph of order n is at most 4n/25. Using flag algebras, Balogh, Clemen, Lidický, Norin and Volec [SIAM J. Discrete Math. 37 (2023) 1173--1179] established the stronger bound 15n/94 for all triangle-free graphs. We investigate the effect of additionally excluding pentagons and prove that every {C_3,C_5}-free graph G of order n satisfies qmin(G)<0.0569n, without any regularity assumption. The proof combines seven-vertex flag inequalities with local Rayleigh constraints that retain the least eigenvalue throughout the counting argument. An exact integer certificate establishes the required inequality.
ER  -