A graph complexity measure based on the spectral analysis of the Laplace operator

In this work we introduce a concept of complexity for undirected graphs in terms the spectral analysis Laplacian operator defined by incidence matrix graph. Precisely, compute norm vector eigenvalues both graph and its complement take their product. Doing so, obtain quantity that satisfies two basic properties are expected measure complexity. First,complexity fully connected disconnected vanish. Second, complementary coincide. This notion allows us to distinguish different kinds placing them "croissant-shaped" region plane link density - complexity, highlighting some features like connectivity,concentration, uniformity or regularity existence clique-like clusters. Indeed, considering with fixed number nodes, plotting versus find generated methods place at regions plane. We consider graphs, particular Erd\"os-R\'enyi, Watts-Strogatz Barab\'asi-Albert models. Also, particular, let say deterministic, wit, lattices, stars, hyper-concentrated cliques-containing graphs. It is worthy noticing these deterministic classical models depict boundary croissant-shaped region. Finally, as an application real measurements, brain connectivity from epileptic patients obtained magnetoencephalography (MEG) recording, baseline period ictal periods .In case, our definition could be used tool discerning between states, differences distinct frequencies MEG recording.