Sharp edge-spectral supersaturation for odd cycles

Abstract

Let \(G\) be a graph with \(m\) edges and adjacency spectral radius\(\rho(G)\), 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 \(\rho(G)\le g_k(m)\). We establish the asymptotically sharp supersaturation counterpart of their result. More precisely, for every fixed integer \(k\ge 2\), we prove that \[ \inf_{\substack{e(G)=m\\ \rho(G)>g_k(m)}} \frac{N(C_{2k+1},G)}{m^k} = \frac{\lceil k^2/2\rceil (k-1)!}{(k+1)^k}+o(1) \qquad\text{as }m\to\infty. \] 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 \( \rho(G)>\frac{1+\sqrt{4m-3}}{2}\) 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.