The spectral radius of graphs without long cycles

Nikiforov conjectured that for a given integer $k\ge 2$, any graph $G$ of sufficiently large order $n$ with spectral radius $\mu(G)\geq \mu(S_{n,k})$ (or $\mu(G)\ge \mu(S_{n,k}^+))$ contains $C_{2k+1}$ or $C_{2k+2}$(or $C_{2k+2}$), unless $G=S_{n,k}$ (or $G=S_{n,k}^+)$, where $C_\ell$ is a cycle of length $\ell$ and $S_{n,k}=K_k\vee \overline{K_{n-k}}$, the join graph of a complete graph of order $k$ and an empty graph on $n-k$ vertices, and $S_{n,k}^+$ is the graph obtained from $S_{n,k}$ by adding an edge in the independent set of $S_{n,k}$. %This can be vie as spectral version of Erd\"{o}s and S\'{o}s conjecture. In this paper, a weaker version of Nikiforov's conjecture is considered, we prove that for a given integer $k\ge 2$, any graph $G$ of sufficiently large order $n$ with spectral radius $\mu(G)\geq \mu(S_{n,k})$ (or $\mu(G)\ge \mu(S_{n,k}^+))$ %$C_{2k+1}$ or $C_{2k+2}$(or $C_{2k+2}$), unless $G=S_{n,k}$ (or $G=S_{n,k}^+)$$S_{n,k}$ ( or $S_{n,k}^+$) is the unique extremal graph with maximum radius among all of the graphs of order $n$ and contains a cycle $C_{\ell}$ with $\ell \geq 2k+1$ (or $C_{\ell}$ with $\ell \geq 2k+2$), unless $G=S_{n,k}$ (or $G=S_{n,k}^+)$. These results also imply a result of Nikiforov given in [Theorem 2, The spectral radius of graphs without paths and cycles of specified length, LAA, 2010].