Optimal tree for both synchronizability and converging time

It has been proved that the spanning tree from a given network has optimal synchronizability, which means the index R = lambda(N)/lambda(2) reaches the minimum 1. Although the optimal synchronizability is corresponding to the minimal critical overall coupling strength to reach synchronization, it does not guarantee a shorter converging time from disorder initial configuration to synchronized state. In this letter, we find that the depth of the tree is the only factor that affects the converging time. The relation between the depth and the converging time is given as well. In addition, we present a simple and universal way to get such an effective oriented tree from a given network to reduce the converging time significantly by minimizing the depth of the tree. The shortest spanning tree has both maximal synchronizability and minimal converging time. Copyright (C) EPLA, 2009