The synchronizability of a connected undirected network is essentially determined by the spectrum of its Laplacian matrix, which reflects most topological characteristics of the network such as degree distribution, shortest-path length, betweenness centrality, among others. Recently, we found that networks with best possible synchronizability are in some sense “homogenous” and “symmetric”, with several common features such as an identical degree sequence, a longest girth, and a shortest path-sum. We have verified this observation by degree-3 regular networks of small sizes, and conjectured this be true in general.