Ming-qing ZHAI, Rui-fang LIU
A set of cycles is called disjoint 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, Erdös and Pósa established the following edge-extremal result: For every graph $G$ of order $n$ which contains no $k$ disjoint 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 Erdös-Pósa Theorem. Let $k\geq1$ and $n\geq \frac{16(2k-1)}{\lambda^{2}}$ with $\lambda=\frac1{120k^2}$. If $G$ is a graph of order $n$ which contains no $k$ disjoint cycles, then $\rho(G)\leq \rho(S_{n,2k-1}),$ the equality holds if and only if $G\cong S_{n,2k-1}.$ From the perspective of counting subgraphs, our result implies that every graph $G$ with $\rho(G)\geq\Theta(n^{\frac35})$ contains at least $\Theta(n^{\frac15})$ disjoint cycles. Finally, a related problem is proposed for further research.
