Fiedler and Nikiforov gave sufficient conditions for the existence ofHamilton paths and cycles in terms of spectral radii of a graph and itscomplement, which stimulated much later works on this topic. The class ofclaw-free graphs, which contains line graphs as a subclass, is an importanttype of graphs. Motivated by Fiedler and Nikiforov's works, in this paper weprove tight sufficient spectral conditions for Hamilton cycles in 2-connectedclaw-free graphs and Hamilton paths in connected claw-free graphs. Our strategyis to deduce spectral results from structural results, and main tools includeRyj\'{a}cek's claw-free closure theory, Tur\'{a}n's theorem and a characterizedtheorem of minimal 2-connected non-hamiltonian claw-free graphs due to Brousek,together with some spectral inequalities.
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