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Graph matching is a fundamental problem in Computer Vision and Machine Learning. We present two contributions. First, we give a new spectral relaxation technique for approximate solutions to matching problems, that naturally incorporates one-to-one or one-to-many constraints within the relaxation scheme. The second is a normalization procedure for existing graph matching scoring functions that can dramatically improve the matching accuracy. It is based on a reinterpretation of the graph matching compatibility matrix as a bipartite graph on edges for which we seek a bistochastic normalization. We evaluate our two contributions on a comprehensive test set of random graph matching problems, as well as on image correspondence problem. Our normalization procedure can be used to improve the performance of many existing graph matching algorithms, including spectral matching, graduated assignment and semidefinite programming.