### Spectral Radius and Traceability of Connected Claw-Free Graphs

#### Abstract

Let $G$ be a connected claw-free graph on $n$ vertices and $\overline{G}$ be its complement graph. Let $\mu(G)$ be the spectral radius of $G$. Denote by $L_{n-3,3}$ the graph consisting of $K_{n-3}$ and three disjoint pendent edges. In this note we prove that: \\

(1) If $\mu(G)\geq n-4$, then $G$ is traceable unless $G=L_{n-3,3}$. \\

(2) If $\mu(\overline{G})\leq \sqrt{3n-8}$ where $n\geq 11$, then $G$ is traceable. \\

Our works are counterparts on claw-free graphs of previous theorems due to Lu et al., and Fiedler and Nikiforov, respectively.

(1) If $\mu(G)\geq n-4$, then $G$ is traceable unless $G=L_{n-3,3}$. \\

(2) If $\mu(\overline{G})\leq \sqrt{3n-8}$ where $n\geq 11$, then $G$ is traceable. \\

Our works are counterparts on claw-free graphs of previous theorems due to Lu et al., and Fiedler and Nikiforov, respectively.

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