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PreprintCC0arXiv

A counterexample to the Fang--Lin conjecture: Edge and spectral extremality diverge near a Turán graph

  • Qi Wu
  • Yong Lu
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Abstract

Fang and Lin [J. Algebraic Combin. 63 (2026), Art.~58] asked whether, whenever F is edge-color-critical with χ(F)=r+1, every non-r-partite, F-free graph of maximum adjacency spectral radius must also maximize the number of edges. We give a negative answer. Let F=K1∨μ(K3), where μ(K3) is the Mycielskian of a triangle. This graph is edge-color-critical with χ(F)=5. We prove that SPEX5(n,F)∩EX5(n,F)=∅ for all sufficiently large n. Thus no graph can simultaneously maximize both the edge count and the spectral radius.

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Cite

BibTeXRIS
BibTeX
@misc{wu2026counterexample,
  title = {{A counterexample to the Fang--Lin conjecture: Edge and spectral extremality diverge near a Turán graph}},
  author = {Qi Wu and Yong Lu},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.37208},
  eprint = {2609.37208},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.37208},
  note = {CC0},
}
RIS
TY  - UNPB
TI  - A counterexample to the Fang--Lin conjecture: Edge and spectral extremality diverge near a Turán graph
AU  - Qi Wu
AU  - Yong Lu
PY  - 2026
DA  - 2026/09/29
PB  - arXiv
DO  - 10.48550/arxiv.2609.37208
UR  - https://arxiv.org/abs/2609.37208
AB  - Fang and Lin [J. Algebraic Combin. 63 (2026), Art.~58] asked whether, whenever F is edge-color-critical with χ(F)=r+1, every non-r-partite, F-free graph of maximum adjacency spectral radius must also maximize the number of edges. We give a negative answer. Let F=K_1∨μ(K_3), where μ(K_3) is the Mycielskian of a triangle. This graph is edge-color-critical with χ(F)=5. We prove that SPEX_5(n,F)∩EX_5(n,F)=∅ for all sufficiently large n. Thus no graph can simultaneously maximize both the edge count and the spectral radius.
ER  -