Suppose we have an undirected weighted graph (more specifically, a scale-free network). When I plot its normalized graph Laplacian spectrum, I get something like this:
Normalized Laplacian Spectrum, multiplicity versus eigenvalues
What is the significance of the eigenvalue corresponding to the highest eigenvalue multiplicity? (which in my case is around 1.15) Is there any intuition behind it?
My question was raised when I saw that this peak approaches to Eigenvalue 1 by randomizing the graph (degree-preserving double-edge swap).