For a $d$-regular graph $G$ with second eigenvalue $\lambda = \lambda_2(A)$: for all vertex sets $S, T \subseteq V$, $\left|e(S, T) - \frac{d|S||T|}{n}\right| \leq \lambda\sqrt{|S||T|}$ where $e(S,T)$ counts edges between $S$ and $T$. When $\lambda \ll d$ (spectral gap is large), edge distribution is close to random: every pair of large sets has approximately the "expected" number of edges.

If $B$ is a principal submatrix of a symmetric $n \times n$ matrix $A$ (obtained by deleting rows and columns), then the eigenvalues interlace: $\lambda_i(A) \geq \lambda_i(B) \geq \lambda_{i+n-m}(A)$ where $B$ is $m \times m$. For graphs: deleting a vertex can decrease each eigenvalue by at most the amount removed. This constrains how the spectrum changes under vertex deletion and is used to prove bounds on clique number, chromatic number, and maximum cut.

A strongly regular graph $\mathrm{srg}(n, k, \lambda, \mu)$ is $k$-regular with every pair of adjacent vertices having $\lambda$ common neighbors and every pair of non-adjacent vertices having $\mu$ common neighbors. The adjacency matrix satisfies $A^2 = kI + \lambda A + \mu(J - I - A)$, giving eigenvalues $k$ (multiplicity 1), $r = \frac{(\lambda-\mu)+\sqrt{(\lambda-\mu)^2+4(k-\mu)}}{2}$, $s = \frac{(\lambda-\mu)-\sqrt{(\lambda-\mu)^2+4(k-\mu)}}{2}$. Feasibility conditions from integrality of multiplicities severely restrict which parameter sets are realizable.