Title: Spectral  Embedding of Regularized Block Models. Abstract: Spectral embedding is a popular technique for the representation of graph data. Several regularization techniques have been proposed to improve the quality of the embedding with respect to downstream tasks like clustering. In this paper, we explain on a simple block model the impact of the complete graph regularization, whereby a constant is added to all entries of the adjacency matrix. Specifically, we show that the regularization forces the spectral embedding  to focus on  the  largest blocks, making the representation less sensitive to noise or outliers. We illustrate these results on both  on both synthetic and real data, showing how regularization improves standard clustering scores. 