AI 中文总结
研究尼基福罗夫关于连续长度圈的开放问题，通过结合度形式的塞梅雷迪正则引理等多种方法，确定了精确常数，改进了此前结果，给出图\(G\)在一定条件下包含特定长度圈的结论。
AI 中文摘要
设\(\rho(G)\)表示\(n\)阶图\(G\)的邻接谱半径。我们确定了尼基福罗夫（2008年）关于连续长度圈的一个开放问题中的精确常数。对于每个\(\varepsilon>0\)和所有足够大的\(n\)，若\(G\)是一个\(n\)顶点图且\(\rho(G)>\sqrt{\lfloor{n^2/4}\rfloor}\)，则\(G\)包含每个整数长度\(3\leq\ell\leq(\frac{3 - \sqrt{5}}{2}-\varepsilon)n\)的圈\(C_{\ell}\)。常数\((3 - \sqrt{5})/2\)是最优的，如分裂图\(K_k\vee\overline{K}_{n - k}\)（其中\(k\sim(3 - \sqrt{5})n/4\)）所示。我们的结果改进了之前所有结果。证明结合了塞梅雷迪正则引理的度形式、冯 - 于 - 张的谱匹配定理、外尔不等式、卢扎克连通匹配嵌入方法的改进以及其他思想。
英文摘要
Let $ρ(G)$ denote the adjacency spectral radius of a graph $G$ of order $n$. We determine the sharp constant in an open problem of Nikiforov (2008) on cycles of consecutive lengths. For every $\varepsilon>0$ and all sufficiently large $n$, if $G$ is an $n$-vertex graph with $ρ(G)>\sqrt{\lfloor{n^2/4}\rfloor},$ then $G$ contains a cycle $C_\ell$ for every integer length $3\le \ell\le (\frac{3-\sqrt5}{2}-\varepsilon)n.$ The constant $(3-\sqrt5)/2$ is best possible, as shown by the split graph $K_k\vee\overline K_{n-k}$ with $k\sim(3-\sqrt5)n/4$. Our result improves all previous results [LAA2008, CPC2020, JGT2023, JGT2023, GC2024]. The proof combines the degree form of Szemerédi's regularity lemma, a spectral matching theorem of Feng-Yu-Zhang, Weyl's inequality, a refinement of Łuczak's connected-matching embedding method, and other ideas.
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