arXiv 2102.038552021-02-07math.CO

连续长度的稳定性方法、特征值和圈

The stability method, eigenvalues and cycles of consecutive lengths

Binlong Li, Bo Ning

备注：: 13 pages

摘要：

Woodall证明了，对于$G$阶为$n\geq 2k+3$的图，其中$k\geq 0$是整数，如果$e(G)\geq \binom{n-k-1}{2}+\binom{k+2}{2}+1$，则$G$对每个$\ell\in [3,n-k]$都包含一个$C_{\ell}$.在本文中，我们证明了该定理的一个稳定性结果。作为一个副产品，我们给出了完整的解决方案，在\cite{GN19}的两个问题。我们的第二部分致力于一个开放的问题Nikiforov：什么是最大的$C$，使所有积极的$\varepsilon<C$和足够大的$n$，每一个图$G$的顺序$n$与频谱半径$ρ(G)>\sqrt{\lfloor\frac{n^2}{4}\rfloor}$包含一个周期的长度$\ell$的每一个$\ell\leq (C-\varepsilon)n$。我们证明了$C\geq\frac{1}{4}$的方法不同于以往的，改进了现有的界限。我们还得到了一个Erdens-Gallai型边数条件的偶数圈，这可能是独立的兴趣。

英文摘要：

Woodall proved that for a graph $G$ of order $n\geq 2k+3$ where $k\geq 0$ is an integer, if $e(G)\geq \binom{n-k-1}{2}+\binom{k+2}{2}+1$ then $G$ contains a $C_{\ell}$ for each $\ell\in [3,n-k]$. In this article, we prove a stability result of this theorem. As a byproduct, we give complete solutions to two problems in \cite{GN19}. Our second part is devoted to an open problem by Nikiforov: what is the maximum $C$ such that for all positive $\varepsilon<C$ and sufficiently large $n$, every graph $G$ of order $n$ with spectral radius $ρ(G)>\sqrt{\lfloor\frac{n^2}{4}\rfloor}$ contains a cycle of length $\ell$ for every $\ell\leq (C-\varepsilon)n$. We prove that $C\geq\frac{1}{4}$ by a method different from previous ones, improving the existing bounds. We also derive an Erdős-Gallai type edge number condition for even cycles, which may be of independent interest.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

摘要：

英文摘要：

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