Sharp edge-spectral supersaturation for odd cycles
Abstract
Let G be a graph with m edges and adjacency spectral radiusρ(G), and let N(C2k+1,G) denote the number of copies of C2k+1 in G. For each fixed integer k≥ 2, define gk(m):=(k−1+√(4m−k2+1))/2. Li, Zhai and Shu [European J. Combin., 2024] determined the spectral extremal threshold for odd cycles by proving that, for all sufficiently large m, every C2k+1-free graph G with m edges satisfies ρ(G)≤ gk(m). We establish the asymptotically sharp supersaturation counterpart of their result. More precisely, for every fixed integer k≥ 2, we prove that infsubstacke(G)=m ρ(G)>gk(m) (N(C2k+1,G))/mk = (⌈ k2/2⌉ (k−1)!)/((k+1)k)+o(1) as m→∞. Thus every m-edge graph whose spectral radius exceeds the C2k+1-free threshold contains at least ( (⌈ k2/2⌉ (k−1)!)/((k+1)k)−o(1) )mk copies of C2k+1, and the leading constant is asymptotically best possible. In particular, taking k=2, we obtain if ρ(G)>(1+√(4m−3))/2 then N(C5,G)≥ (2/9−o(1))m2, with the constant 2/9 being asymptotically optimal. This answers a question of Chen, Li and Tang concerning the existence and the largest possible value of a constant C>0 for which the same spectral condition guarantees at least Cm2 copies of C5. More generally, our result resolves a recent problem of Li, Lin, Liu and Zhang on spectral supersaturation for odd cycles. The proof combines spectral stability and resolvent analysis with estimates for odd spectral moments and a careful treatment of non-injective closed walks.
Related
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- Optimal spectral supersaturation for cliques and odd cycles
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- Power-of-two cycles in projective cubic graphs of large edge-width
BibTeX
@misc{liu2026sharp,
title = {{Sharp edge-spectral supersaturation for odd cycles}},
author = {Jiaqi Liu and Zhenzhen Lou and Shuang Sun},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.08077},
eprint = {2609.08077},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.08077},
}
RIS
TY - UNPB
TI - Sharp edge-spectral supersaturation for odd cycles
AU - Jiaqi Liu
AU - Zhenzhen Lou
AU - Shuang Sun
PY - 2026
DA - 2026/09/08
PB - arXiv
DO - 10.48550/arxiv.2609.08077
UR - https://arxiv.org/abs/2609.08077
AB - Let G be a graph with m edges and adjacency spectral radiusρ(G), and let N(C_2k+1,G) denote the number of copies of C_2k+1 in G. For each fixed integer k≥ 2, define g_k(m):=(k−1+√(4m−k^2+1))/2. Li, Zhai and Shu [European J. Combin., 2024] determined the spectral extremal threshold for odd cycles by proving that, for all sufficiently large m, every C_2k+1-free graph G with m edges satisfies ρ(G)≤ g_k(m). We establish the asymptotically sharp supersaturation counterpart of their result. More precisely, for every fixed integer k≥ 2, we prove that inf_substacke(G)=m ρ(G)>g_k(m) (N(C_2k+1,G))/m^k = (⌈ k^2/2⌉ (k−1)!)/((k+1)^k)+o(1) as m→∞. Thus every m-edge graph whose spectral radius exceeds the C_2k+1-free threshold contains at least ( (⌈ k^2/2⌉ (k−1)!)/((k+1)^k)−o(1) )m^k copies of C_2k+1, and the leading constant is asymptotically best possible. In particular, taking k=2, we obtain if ρ(G)>(1+√(4m−3))/2 then N(C_5,G)≥ (2/9−o(1))m^2, with the constant 2/9 being asymptotically optimal. This answers a question of Chen, Li and Tang concerning the existence and the largest possible value of a constant C>0 for which the same spectral condition guarantees at least Cm^2 copies of C_5. More generally, our result resolves a recent problem of Li, Lin, Liu and Zhang on spectral supersaturation for odd cycles. The proof combines spectral stability and resolvent analysis with estimates for odd spectral moments and a careful treatment of non-injective closed walks.
ER -