Spectral radius and Hamiltonicity of graphs

Let G be a graph of order n and μ(G) be the largest eigenvalue of its adjacency matrix. Let be the complement of G.Write Kn-1+v for the complete graph on n-1 vertices together with an isolated vertex, and Kn-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=Kn-1+v; if strict inequality holds, then G contains a Hamiltonian cycle unless G=Kn-1+e.If , then G contains a Hamiltonian path unless G=Kn-1+v.If , then G contains a Hamiltonian cycle unless G=Kn-1+e.