  Let $G$ be a graph with minimum degree $\delta$. The spectral radius of $G$,
denoted by $\rho(G)$, is the largest eigenvalue of the adjacency matrix of $G$.
In this note we mainly prove the following two results. (1) Let $G$ be a graph
on $n\geq 4$ vertices with $\delta\geq 1$. If $\rho(G)> n-3$, then $G$ contains
a Hamilton path unless $G\in\{K_1\vee (K_{n-3}+2K_1),K_2\vee 4K_1,K_1\vee
(K_{1,3}+K_1)\}$. (2) Let $G$ be a graph on $n\geq 14$ vertices with $\delta
\geq 2$. If $\rho(G)\geq \rho(K_2\vee (K_{n-4}+2K_1))$, then $G$ contains a
Hamilton cycle unless $G= K_2\vee (K_{n-4}+2K_1)$. As corollaries of our first
result, two previous theorems due to Fiedler and Nikiforov and Lu et al. are
obtained, respectively. Our second result refines another previous theorem of
Fiedler and Nikiforov.
