论文标题
光谱erdős-pósa定理
A spectral Erdős-Pósa Theorem
论文作者
论文摘要
如果其中两个没有一个共同的顶点，则一组循环被称为独立。令$ s_ {n，2k-1} $为完整的拆分图，这是一个大小$ 2K-1 $的联接，带有独立尺寸$ n-2k+1 $的独立集。 1962年，Erdős和Pósa建立了以下边缘效果：对于每个图$ g $的订单$ n $，其中包含no $ k $独立循环，其中$ k \ geq2 $和$ n \ geq 24k $，我们只有$ e（g）\ e（g）\ leq（2k-1）（n-k），$ g and quangity，如果是$ g}，如果论文，我们证明了Erdős-Pósa定理的光谱版本。令$ k \ geq1 $和$ n \ geq \ frac {16（2k-1）} {λ^{2}} $带有$λ= \ frac1 {120k^2} $。如果$ g $是订单$ n $的图表，其中包含no $ k $独立周期，则$ρ（g）\ leqρ（s_ {n，2k-1}），$，$仅在且仅当$ g \ g \ cong s_ {n，2k-1}时才能保持平等。这是一个新的示例示例示例示例，以示出了一个新的expration for Edge-expremal问题。最后，提出了一个相关的问题以进行进一步研究。
A set of cycles is called independent if no two of them have a common vertex. Let $S_{n, 2k-1}$ be the complete split graph, which is the join of a clique of size $2k-1$ with an independent set of size $n-2k+1$. In 1962, Erdős and Pósa established the following edge-extremal result: for every graph $G$ of order $n$ which contains no $k$ independent cycles, where $k\geq2$ and $n\geq 24k$, we have $e(G)\leq (2k-1)(n-k),$ with equality if and only if $G\cong S_{n,2k-1}.$ In this paper, we prove a spectral version of Erdős-Pósa Theorem. Let $k\geq1$ and $n\geq \frac{16(2k-1)}{λ^{2}}$ with $λ=\frac1{120k^2}$. If $G$ is a graph of order $n$ which contains no $k$ independent cycles, then $ρ(G)\leq ρ(S_{n,2k-1}),$ the equality holds if and only if $G\cong S_{n,2k-1}.$ This presents a new example illustration for which edge-extremal problems have spectral analogues. Finally, a related problem is proposed for further research.
