A complete characterization of graphs with second largest eigenvalue at most (√5−1)/2

@misc{chen2026complete,
  title = {{A complete characterization of graphs with second largest eigenvalue at most $(\sqrt5-1)/2$}},
  author = {Chaohui Chen and Hangxi Cha and Haiying Shan and Muhuo Liu},
  howpublished = {arXiv},
  year = {2026},
  month = nov,
  doi = {10.48550/arxiv.2611.02005},
  eprint = {2611.02005},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2611.02005},
}

TY  - UNPB
TI  - A complete characterization of graphs with second largest eigenvalue at most (√5−1)/2
AU  - Chaohui Chen
AU  - Hangxi Cha
AU  - Haiying Shan
AU  - Muhuo Liu
PY  - 2026
DA  - 2026/11/01
PB  - arXiv
DO  - 10.48550/arxiv.2611.02005
UR  - https://arxiv.org/abs/2611.02005
AB  - In 1993, Cao and Hong posed the problem of characterizing graphs whose second largest adjacency eigenvalue is greater than 1/3 and less than (√5−1)/2. Subsequent work extended the question to the non-strict bound. We give a complete structural characterization of connected graphs G with |V(G)|≥2 satisfying λ_2(G)≤(√5−1)/2. Their complements are described by disjoint unions of graphs from an explicit family with least adjacency eigenvalue at least −(1+√5)/2, possibly together with a cone or a thin or thick spider constructed from disjoint unions of members of the same family, subject to explicit numerical inequalities. This provides a positive characterization of the class, going beyond the classical result of Cvetković and Simić that the bound λ_2≤(√5−1)/2 admits only finitely many minimal forbidden induced subgraphs.
ER  -