c\right) \geq 0$, for $j=1,2,$ and $3$. Similarly, all the eigenvalues are nonnegative if and only if $p_{2}\left( c\right) \geq 0$ and $p_{1}\left( c\right) ,p_{3}\left( c\right) \leq 0$. In fact, one can prove that the optimal value of $\alpha _{2}$ and $\beta _{2}$ in (\[Bound max eigenval\]) can be computed as the smallest and the largest roots of $\det
H_{2}\left( c\right) =0$, which yields a third degree polynomial in the variable $c$ [@Las11]. There are closed-form expressions for the roots of this polynomial (for example, Cardano’s formula [@AS65]), although the resulting expressions for the roots are rather complicated.

In this subsection, we have presented a convex optimization framework to compute optimal bounds on the maximum and the minimum eigenvalues of a graph $\mathcal{G}$ from a truncated sequence of its spectral moments. Since we have expressions for spectral moments in terms of local structural properties, these bounds relate the eigenvalues of a graph with its structural properties.

\[Simulations\]Numerical Simulations
====================================

As we illustrated in Section \[Strategic Section\], there is a close connection between the largest and the smallest eigenvalues of a network and the outcome of a game with linear best response functions. In this section, we use our bounds on 