We give some inequalities relating the number of edges that two graphs of the same order may have in common with their degree sequences and their spectra. We then give some examples and related inequalities, and present applications to independence number and coloration.

Two graphs are co-spectral if their respective adjacency matrices have the same multi-set of eigenvalues. A graph is said to be determined by its spectrum if all graphs that are co-spectral with it are isomorphic to it. We consider these properties in relation to logical definability. We show that any pair of graphs that are elementarily equivalent with respect to the three-variable counting fi...