Eigenvalues of the adjacency matrix reveal network structure. Random graphs → Wigner semicircle. Scale-free → heavy tail spike. Regular → discrete bands.
Network (nodes colored by leading eigenvector)
Spectral density ρ(λ)
The spectrum of A encodes: connectivity (λ₁), bipartiteness (symmetric spectrum), graph diameter, chromatic number bounds. For an Erdős-Rényi G(n,p) random graph, eigenvalues follow Wigner's semicircle law as n→∞. Scale-free networks develop a large isolated λ₁ corresponding to the hub, plus a bulk continuum.