# Spectral Graph Theory [closed]

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 GodsilMay 13 '14 at 0:54

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• What do you already know about this? From what background are you coming at the problem? This will help potential answerers. – David Roberts Apr 12 '14 at 11:39
• We already have an adjacency spectrum then what is the advantage of laplacian spectrum over adjacency spectrum and as these two matrices are related, there must exist some relationship between their spectra right? – user49525 Apr 14 '14 at 4:36
• "there must exist some relationship between their spectra right?" Well, why do you think there is a relation between the spectra of the sum of two matrices and the spectra of the individual matrices? (Some things can be said for Hermitian matrices, but they are probably harder and deeper and less precise than what you seem to be hoping for.) – Yemon Choi May 12 '14 at 23:56

• I offered only the "the short answer". For example, if the graph has one vertex of degree $k$ and the rest have degree $\ell$, you can derive an interlacing result. But I have never seen anyone find a useful connection between the two spectra. – Chris Godsil Apr 12 '14 at 17:50