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In this paper, the relationship between the network synchronizability and theedge distribution of its associated graph is investigated. First, it is shownthat adding one edge to a cycle definitely decreases the networksychronizability. Then, since sometimes the synchronizability can be enhancedby changing the network structure, the question of whether the networks withmore edges are easier to synchronize is addressed. It is shown by examples thatthe answer is negative. This reveals that generally there are redundant edgesin a network, which not only make no contributions to synchronization butactually may reduce the synchronizability. Moreover, an example shows that thenode betweenness centrality is not always a good indicator for the networksynchronizability. Finally, some more examples are presented to illustrate howthe network synchronizability varies following the addition of edges, where allthe examples show that the network synchronizability globally increases butlocally fluctuates as the number of added edges increases.

Are networks with more edges easier to synchronize?