\right) \succcurlyeq 0\right\} ,  \label{Bound max eigenval}\end{aligned}$$for $\left[ a,b\right] $ being the smallest interval containing $supp(\mu )$.

Observe that $\alpha _{s}$ and $\beta _{s}$ are the solutions to two SDP’s in one variable, since the constraint $H_{s}\left( \alpha \right)
\succcurlyeq 0$ (resp. $-H_{s}\left( \beta \right) \succcurlyeq 0$) indicates that the matrix $H_{s}\left( \alpha \right) $ (resp. $-H_{s}\left(
\beta \right) $) is positive semidefinite and this matrix has affine entries with respect to $\alpha $ (resp. $\beta $). Hence, they can be efficiently computed using standard optimization software (for example, CVX [@CVX]). As we increase $s$ in Proposition \[Lasserre bound\], more moments are involved in the SDP’s, and the resulting bounds become tighter, i.e., $%
\alpha _{s+1}\leq \alpha _{s}$ and $\beta _{s+1}\geq \beta _{s}$.

In the context of our spectral analysis, the Borel measure in Proposition \[Lasserre bound\] corresponds to the spectral density of a graph $%
\mathcal{G}$, and the smallest interval $\left[ a,b\right] $ corresponds to $%
\left[ \lambda _{n}\left( A_{\mathcal{G}}\right) ,\rho \left( A_{\mathcal{G}%
}\right) \right] $. Thus, Proposition \[Lasserre bound\] provides an efficient numerical scheme to compute the bounds $\alpha _{s}\geq \lambda
_{n}\left( A_{\mathcal{G}}\right) $ and $\beta _{s}\leq \rho \left( A_{%
\mathcal{G}}\right) $. When we are given a sequence 