Generally, synchronization in a network of chaotic systems depends on the underlying coupling topology. Recently, there have been several studies conducted to determine what features of this topology contribute to the ability to synchronize. A short diameter has been proposed by several authors as a means to facilitate synchronization whereas others point to features such as the homogeneity of the degree sequence. Recently, it has been shown that the degree sequence by itself is not sufficient to determine synchronizability. The purpose of this Letter is to continue this study. For a given degree sequence, we construct two connected graphs with this degree sequence whose synchronizability can be quite different. In particular, we construct a graph with low synchronizability which improves upon previous bounds under certain conditions and we construct a graph which has synchronizability that is asymptotically maximal. On the other hand, we show analytically that for a random network model, homogeneity of the degree sequence is beneficial to synchronization.