  This paper presents sufficient conditions for Hamiltonian paths and cycles in
graphs. Letting $\lambda\left( G\right) $ denote the spectral radius of the
adjacency matrix of a graph $G,$ the main results of the paper are:
  (1) Let $k\geq1,$ $n\geq k^{3}/2+k+4,$ and let $G$ be a graph of order $n$,
with minimum degree $\delta\left( G\right) \geq k.$ If \[ \lambda\left(
G\right) \geq n-k-1, \] then $G$ has a Hamiltonian cycle, unless
$G=K_{1}\vee(K_{n-k-1}+K_{k})$ or $G=K_{k}\vee(K_{n-2k}+\overline{K}_{k})$.
  (2) Let $k\geq1,$ $n\geq k^{3}/2+k^{2}/2+k+5,$ and let $G$ be a graph of
order $n$, with minimum degree $\delta\left( G\right) \geq k.$ If \[
\lambda\left( G\right) \geq n-k-2, \] then $G$ has a Hamiltonian path, unless
$G=K_{k}\vee(K_{n-2k-1}+\overline {K}_{k+1})$ or $G=K_{n-k-1}+K_{k+1}$
  In addition, it is shown that in the above statements, the bounds on $n$ are
tight within an additive term not exceeding $2$.
