Quadratic inequalities between the largest eigenvalues of a graph

We prove a sharp quadratic inequality between the largest two eigenvalues ${\lambda }_1\ge {\lambda }_2$ of a graph with $n$ vertices. We also prove a quadratic inequality between the second and third largest eigenvalues ${\lambda }_2\ge {\lambda }_3$. These results in particular imply the bounds ${\lambda }_1+{\lambda }_2\le \frac{8}{7}n-2$, ${\lambda }_3\le \frac{n}{3}-1$ and ${\lambda }_2+{\lambda }_3\le \frac{2}{3}n-2$. In fact we determine the closure of the set of possible $(\frac{{\lambda }_1+1}{n},\frac{{\lambda }_2+1}{n})\in {\mathbb{R}}^2$ and $(\frac{{\lambda }_2+1}{n},\frac{{\lambda }_3+1}{n})\in {\mathbb{R}}^2$. More generally, we prove quadratic bounds in the case of symmetric matrices in $[0,1{]}^{n\times n}$, and we also give a quadratic bound for two eigenvalues of a symmetric matrix in $[-1,1{]}^{n\times n}$. These bounds are proved by transforming the problem into extremal geometric questions in ${\mathbb{R}}^3$ and ${\mathbb{R}}^2$. We use the method of Lagrange multipliers to reduce to special cases with at most five points, and we deal with these special cases directly.