Does the eigenratio λ 2/λ N represent the synchronizability of a complex network?

In the study of complex networks, it is commonly believed that the eigenratio λ ₂/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 λ ₂/λ _N does represent the network synchronizability. © 2012 Chinese Physical Society and IOP Publishing Ltd.