Does the eigenratio λ₂/λ_N represent the synchronizability of a complex network?

In the study of complex networks, it is commonly believed that the eigenratio λ₂/λ_N of the Laplacian matrix of a network represents the network synchronizability, especially for symmetric networks. This paper gives two counterexamples to show that this is not true for the case where the network has a disconnected synchronized region. Consequently, a simple answer is presented to the question of when the eigenratio λ₂/λ_N does represent the network synchronizability.