This paper presents sufficient conditions for Hamiltonian paths and cycles in graphs. Letting $lambdaleft( Gright) $ denote the spectral radius of the adjacency matrix of a graph $G,$ the main results of the paper are: (1) Let $kgeq1,$ $ngeq k^{3}/2+k+4,$ and let $G$ be a graph of order $n$, with minimum degree $deltaleft( Gright) geq k.$ If [ lambdaleft( Gright) 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 $kgeq1,$ $ngeq k^{3}/2+k^{2}/2+k+5,$ and let $G$ be a graph of order $n$, with minimum degree $deltaleft( Gright) geq k.$ If [ lambdaleft( Gright) 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 $2$.
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