We study the synchronization of identical oscillators diffusively coupled through a network and examine how adding, removing, and moving single edges affects the ability of the network to synchronize. We present algorithms which use methods based on node degrees and based on spectral properties of the network Laplacian for choosing edges that most impact synchronization. We show that rewiring based on the network Laplacian eigenvectors is more effective at enabling synchronization than methods based on node degree for many standard network models. We find an algebraic relationship between the eigenstructure before and after adding an edge and describe an efficient algorithm for computing Laplacian eigenvalues and eigenvectors that uses the network or its complement depending on which is more sparse.

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The Laplacian L=DA, where D is a diagonal degree matrix and A is the adjacency matrix for the network. Alternatively the normalized Laplacian L=ID12LD12 is sometimes used. In both cases, networks with weighted edges can be considered. Appropriate adjustments to our analysis work for these models of connectivity and produce similar results.

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We denote the sorted Laplacian eigenvalues 0=λ1λ2λn and corresponding eigenvectors v1,v2,vN. The eigenvalue λ1=0 corresponds to spatially uniform perturbations v1=[1,,1].

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