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Asymptotic Spectral Bounds under Odd-Girth Constraints

  • Chen Wang
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Abstract

Odd girth and the sum of the largest and smallest adjacency eigenvalues are two common indicators of how close a graph is to being bipartite. This paper investigates the asymptotic behavior of this eigenvalue sum under odd-girth constraints. We study the associated abstract moment problem over finite real lists satisfying the vanishing odd-moment identities imposed by odd girth at least g and the standard second-moment bound for graph spectra. We prove that the supremum of (λ1+λn)/n over these lists lies between (1−o(1))(log g)2/g3 and (10/7+o(1))(log g)2/g3 as g→∞ through the odd integers. The upper bound applies in particular to graphs of odd girth at least g. For regular graphs of odd girth g≥5, we obtain a stronger bound (λ1+λn)/n ≤ 12/((g−1)(g−2)(g−3)), which is optimal up to a constant factor.

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Cite

BibTeXRIS
BibTeX
@misc{wang2026asymptotic,
  title = {{Asymptotic Spectral Bounds under Odd-Girth Constraints}},
  author = {Chen Wang},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.05172},
  eprint = {2610.05172},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.05172},
}
RIS
TY  - UNPB
TI  - Asymptotic Spectral Bounds under Odd-Girth Constraints
AU  - Chen Wang
PY  - 2026
DA  - 2026/10/04
PB  - arXiv
DO  - 10.48550/arxiv.2610.05172
UR  - https://arxiv.org/abs/2610.05172
AB  - Odd girth and the sum of the largest and smallest adjacency eigenvalues are two common indicators of how close a graph is to being bipartite. This paper investigates the asymptotic behavior of this eigenvalue sum under odd-girth constraints. We study the associated abstract moment problem over finite real lists satisfying the vanishing odd-moment identities imposed by odd girth at least g and the standard second-moment bound for graph spectra. We prove that the supremum of (λ_1+λ_n)/n over these lists lies between (1−o(1))(log g)^2/g^3 and (10/7+o(1))(log g)^2/g^3 as g→∞ through the odd integers. The upper bound applies in particular to graphs of odd girth at least g. For regular graphs of odd girth g≥5, we obtain a stronger bound (λ_1+λ_n)/n ≤ 12/((g−1)(g−2)(g−3)), which is optimal up to a constant factor.
ER  -