Let $λ_{1}(G)$ and $μ_{1}(G)$ denote the spectral radius and the Laplacian spectral radius of a graph $G$, respectively. Li in [Electronic J. Linear Algebra 34 (2018) 389-392] proved sharp upper bounds of $λ_{1}(G)$ based on the connectivity to assure a connected graph to be Hamiltonian and traceable, respectively. In this paper, we present best possible upper bounds of $λ_{1}(G)$ for $k$-connected graphs to be Hamiltonian-connected and homogeneously traceable, respectively. Furthermore, best possible upper bounds of $μ_{1}(G)$ to predict $k$-connected graphs to be Hamiltonian-connected, Hamiltonian and traceable are originally proved, respectively.