奇圈的锐边谱超饱和性
Sharp edge-spectral supersaturation for odd cycles
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中文总结 AI 辅助
该研究证明，对于每个固定整数k≥2，谱半径超过奇圈自由阈值的m边图至少包含渐近最优数量的C_{2k+1}副本，解决了奇圈谱超饱和性问题。
中文摘要 AI 辅助
设 \\(G\\) 是一个具有 \\(m\\) 条边和邻接谱半径 \\(\rho(G)\\) 的图，并设 \\(N(C_{2k+1},G)\\) 表示 \\(C_{2k+1}\\) 在 \\(G\\) 中的副本数。对于每个固定整数 \\(k\ge 2\\)，定义 \\( g_k(m):=\frac{k-1+\sqrt{4m-k^2+1}}{2}. \\) Li、Zhai 和 Shu [European J. Combin., 2024] 通过证明对于所有足够大的 \\(m\\)，每个具有 \\(m\\) 条边的 \\(C_{2k+1}\\)-自由图 \\(G\\) 满足 \\(\rho(G)\le g_k(m)\\)，确定了奇圈的谱极值阈值。我们建立了他们结果的渐近锐利的超饱和性对应部分。更精确地说，对于每个固定整数 \\(k\ge 2\\)，我们证明 \\[ \inf_{\substack{e(G)=m\\\\ \rho(G)>g_k(m)}} \frac{N(C_{2k+1},G)}{m^k} = \frac{\lceil k^2/2\rceil (k-1)!}{(k+1)^k}+o(1) \qquad\text{当 }m\to\infty. \\] 因此，每个谱半径超过 \\(C_{2k+1}\\)-自由阈值的 \\(m\\)-边图至少包含 \\[ \Big( \frac{\lceil k^2/2\rceil (k-1)!}{(k+1)^k}-o(1) \Big)m^k \\] 个 \\(C_{2k+1}\\) 副本，且该首项常数是渐近最优的。特别地，取 \\(k=2\\)，我们得到如果 \\( \rho(G)>\frac{1+\sqrt{4m-3}}{2}\\) 则 \\( N(C_5,G)\ge \left(\frac{2}{9}-o(1)\right)m^2, \\) 其中常数 \\(2/9\\) 是渐近最优的。这回答了 Chen、Li 和 Tang 关于存在性和最大可能值的问题，即是否存在常数 \\(C>0\\) 使得相同的谱条件保证至少 \\(Cm^2\\) 个 \\(C_5\\) 副本。更一般地，我们的结果解决了 Li、Lin、Liu 和 Zhang 最近关于奇圈谱超饱和性的问题。证明结合了谱稳定性和预解分析，以及奇谱矩的估计和对非单射闭游走的仔细处理。
英文摘要
Let \(G\) be a graph with \(m\) edges and adjacency spectral radius\(ρ(G)\), and let \(N(C_{2k+1},G)\) denote the number of copies of \(C_{2k+1}\) in \(G\). For each fixed integer \(k\ge 2\), define \( g_k(m):=\frac{k-1+\sqrt{4m-k^2+1}}{2}. \) 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 \(ρ(G)\le g_k(m)\). We establish the asymptotically sharp supersaturation counterpart of their result. More precisely, for every fixed integer \(k\ge 2\), we prove that \[ \inf_{\substack{e(G)=m\\ ρ(G)>g_k(m)}} \frac{N(C_{2k+1},G)}{m^k} = \frac{\lceil k^2/2\rceil (k-1)!}{(k+1)^k}+o(1) \qquad\text{as }m\to\infty. \] Thus every \(m\)-edge graph whose spectral radius exceeds the \(C_{2k+1}\)-free threshold contains at least \[ \Big( \frac{\lceil k^2/2\rceil (k-1)!}{(k+1)^k}-o(1) \Big)m^k \] copies of \(C_{2k+1}\), and the leading constant is asymptotically best possible. In particular, taking \(k=2\), we obtain if \( ρ(G)>\frac{1+\sqrt{4m-3}}{2}\) then \( N(C_5,G)\ge \left(\frac{2}{9}-o(1)\right)m^2, \) 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.
发表机构
- University of Shanghai for Science and Technology（上海理工大学）
- Institute for Basic Science (IBS)（韩国基础科学研究院）
- Shanghai Jiao Tong University（上海交通大学）
机构由 AI 辅助整理，请以论文原文为准。
