AI 中文总结
本文用谱盈余衡量图超出诺萨尔阈值程度，证明了关于三角形、书本、风筝等的边谱超饱和结果，如特定条件下图含三角形数量、书本大小、风筝数量等，改进了相关界并确定了渐近常数。
AI 中文摘要
在本文中，我们使用谱盈余\(\lambda(G)-\sqrt{m}\)来衡量图\(G\)超出诺萨尔阈值的程度，并证明了关于三角形和书本的以下边谱超饱和结果。(a) 每个具有\(m\geq3\)条边且\(\lambda(G)\geq1 + \sqrt{m - 2}\)的图\(G\)至少包含\(m - 2\)个三角形，当且仅当\(G = K_3\vee\tfrac{m - 3}{3}K_1\)时取等号。(b) 每个\(m\)条边的图\(G\)满足\(t(G)\geq m(\lambda - \sqrt{m})\)，当且仅当\(G\)是完全二分图时取等号。(c) 每个\(m\)条边的诺萨尔图\(G\)包含大小大于\(\tfrac{1}{4}\sqrt{m}\)的书本。(d) 每个\(m\)条边的诺萨尔图\(G\)包含至少\((\tfrac{1}{8}-o(1))m\)个风筝\(C_4^+=B_2\)，且常数\(\tfrac{1}{8}\)是最优的。
英文摘要
In this paper, we use the spectral surplus $λ(G) - \sqrt{m}$ to measure how far $G$ lies above the Nosal threshold, and prove the following edge-spectral supersaturation results for triangles and books. (a) Every graph $G$ with $m\ge 3$ edges and $λ(G) \ge 1 + \sqrt{m-2}$ contains at least $m-2$ triangles, with equality if and only if $G = K_3 \vee \tfrac{m-3}{3} K_1$. This can be viewed as the third-layer supersaturation in the jump phenomenon, after the first layer $t(G) \ge \lfloor \tfrac{1}{2}(\sqrt{m}-1) \rfloor$ proved by Ning and Zhai, and the second layer $t(G) \ge \tfrac{m-1}{2}$ by Zhang and Zhai. (b) Every $m$-edge graph $G$ satisfies $t(G) \ge m\bigl(λ- \sqrt{m}\,\bigr)$, with equality if and only if $G$ is complete bipartite. Consequently, $λ(G) \ge \sqrt{m} + q$ forces $t(G) > q m$ for every real $q > 0$. This is an edge-spectral counterpart of the Lovász--Simonovits theorem, and it improves the Bollobás--Nikiforov bound $t(G) \ge \tfrac13 λ(λ^2 - m)$ in the range $\sqrt m \le λ(G) \le 1.3\sqrt m $. (c) Every $m$-edge Nosal graph $G$ contains a book of size greater than $\tfrac14 \sqrt{m}$. This improves two recent results on the booksize constant: $\tfrac{1}{24}$ proved by Li, Liu and Zhang, and $\tfrac19$ by Zhai, Li and Lou. This narrows the gap toward the conjectured optimal constant $\tfrac13$. (d) Every $m$-edge Nosal graph $G$ contains at least $\bigl(\tfrac{1}{8} - o(1)\bigr) m$ copies of the kite $C_4^+=B_2$, and the constant $\tfrac18$ is best possible. This determines the sharp asymptotic constant for counting $C_4^+$ and strengthens the $Ω(m)$ bound of Li, Liu and Zhang.
Comments31 pages, 1 figure. Spectral extremal graph theory
