Let G be an undirected graph, then Laplacian Matrix(L(G)) = Degree Matrix (D(G)) - Adjacency Matrix (A(G)). What is the relationship between laplacian and adjacency spectrum of undirected graphs?
closed as too broad by Matthew Kahle, Stefan Kohl, Misha, Yemon Choi, Chris Godsil May 13 '14 at 0:54
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The short answer is that, if your graph is not regular, there is no relation. The effect on the eigenvalues of adding a diagonal matrix is the same as adding an arbitary symmetric matrix.