A Spectral Turán Theorem

If all nonzero eigenvalues of the (normalized) Laplacian of a graph $G$ are close to 1, then $G$ is $t$-Turán in the sense that any subgraph of $G$ containing no $K_{t+1}$ contains at most $(1-1/t + o(1) ) e(G)$ edges where $e(G)$ denotes the number of edges in G.