Spectral radius and Hamiltonicity of graphs

Abstract Let G be a graph of order n and μ ( G ) be the largest eigenvalue of its adjacency matrix. Let G ¯ be the complement of G . Write K n - 1 + v for the complete graph on n - 1 vertices together with an isolated vertex, and K n - 1 + e for the complete graph on n - 1 vertices with a pendent edge. We show that: If μ ( G ) ⩾ n - 2 , then G contains a Hamiltonian path unless G = K n - 1 + v ; if strict inequality holds, then G contains a Hamiltonian cycle unless G = K n - 1 + e . If μ ( G ¯ ) ⩽ n - 1 , then G contains a Hamiltonian path unless G = K n - 1 + v . If μ ( G ¯ ) ⩽ n - 2 , then G contains a Hamiltonian cycle unless G = K n - 1 + e .