In general, L+A=D, where D is the diagonal matrix of vertex degrees, so one can use Weyl's Theorem to produce all kinds of inequalities involving both kinds of spectra. There are more refined results too but they are all built around this idea, afaik. Have a look at this paper by a master at the art, for instance:
https://arxiv.org/abs/1607.03015
If you wish to discuss a specific idea or question pertaining to this, feel free to contact me directly.
If every vertex in the graph has the same degree d, then the Laplacian matrix is L=d*I-A
where I is the identity matrix and A is the adjacency matrix. If t1,...,tn are the eigenvalues of A, then d-t1,...,d-tn are the eigenvalues of L. If not all vertices of the graph have the same degree, then I don't know.
In general, L+A=D, where D is the diagonal matrix of vertex degrees, so one can use Weyl's Theorem to produce all kinds of inequalities involving both kinds of spectra. There are more refined results too but they are all built around this idea, afaik. Have a look at this paper by a master at the art, for instance:
https://arxiv.org/abs/1607.03015
If you wish to discuss a specific idea or question pertaining to this, feel free to contact me directly.
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