Is there any relationship between eigenvalues(or spectrum) of graph Laplacian matrix and the eigenvalues of the product of a real symmetric matrix and the Laplacian matrix?
My problem at hand is as follows :
Let A=L*B.
What is the relationship between spectrum (or eigenvalues) of L with the spectrum of A?
L is Laplacian of an undirected graph, hence real symmetric and singular. B is a real symmetric matrix.
I want to show that if I increase the magnitude of eigenvalues of L, the eigenvalues of A will also increase. However, all I could find was a trace inequality relationship, and inequality doesn't necessarily lead to any conclusion.