I'm not a mathematician and I have a question about spectral graph theory.
Is it possible to conclude that we have a fully connected network, if an adjacency spectra of a graph is continuous with no distinct peaks in eigenvalues and a graph does not have any kind of basic, regular graph shape (not a star, path, regular, tree type of a graph)? And if spectra is continous, but contains some kind of peaks in eigenvalues, then the graph is not fully connected network, but rather a collection of distinct clusters?
Thanks in advance for your answers!