The spectral radius of graphs without long cycles

Abstract

Nikiforov conjectured that for a given integer $k\ge 2$, any graph $G$ of sufficiently large order $n$ with spectral radius $\mu (G)\ge \mu (S_{n,k})$ (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 Erdös and Sós 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 $\mu (G)\ge \mu (S_{n,k})$ (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 $\ell \ge 2k+1$ (or $C_{\ell }$ with $\ell \ge 2k+2$), 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].

17 pages