The network structure of optimal synchronizability for bounded regions case

To study the network structure of optimal synchronizability with given node and edge number, we use simple graph to analyze the topology structure of the optimal synchronizability network. Then, the optimal network characteristic of N node N-1 edges, N edges, or N+1 edges are founded. Espectively, An approach to construct the optimal synchronizability network with N nodes and N+1 edges is presented for given node and edge number. And, it is found that the class optimal synchronizability network structure with three chains. Every chain size is as same as possible, that is, maximum and minimum chain the difference between the number of nodes is less than or equal to 1. Moreover, the difference between maximum degree and minimum degree is less than or equal to 1 in the network with N+1 edges. Then, an example is given to illustration the process of construction optimal network with N node and N+1 edges. At last, we discuss on the optimal network with N node and N+2 edges.