Quadratic inequalities between the largest eigenvalues of a graph

We prove a sharp quadratic inequality between the largest two eigenvalues $λ_1 \ge λ_2$ of a graph with $n$ vertices. We also prove a quadratic inequality between the second and third largest eigenvalues $λ_2 \ge λ_3$. These results in particular imply the bounds $λ_1 + λ_2 \le \frac{8}{7} n - 2$, $