need to introduce a probabilistic interpretation of a network eigenvalue spectrum and its spectral moments. For a simple graph $\mathcal{G}$, we define its spectral density as:$$\mu _{\mathcal{G}}\left( x\right) =\frac{1}{n}\sum_{i=1}^{n}\delta \left(
x-\lambda _{i}\right) ,  \label{Spectral Measure}$$where $\delta \left( \bullet \right) $ is the Dirac delta function and $%
\left\{ \lambda _{i}\right\} _{i=1}^{n}$ is the set of (real) eigenvalues of the (symmetric) adjacency matrix $A_{\mathcal{G}}$. Consider a random variable $X$ with probability density $\mu _{\mathcal{G}}$. The moments of $%
X\sim \mu _{\mathcal{G}}$ are equal to the spectral moments of $A_{\mathcal{G%
}}$, i.e.,$$\begin{aligned}
\mathbb{E}_{\mu _{\mathcal{G}}}\left( X^{k}\right) & =\int_{\mathbb{R}%
}x^{k}\mu _{\mathcal{G}}\left( x\right) dx \\
& =\frac{1}{n}\sum_{i=1}^{n}\int_{\mathbb{R}}x^{k}\delta \left( x-\lambda
_{i}\right) dx \\
& =\frac{1}{n}\sum_{i=1}^{n}\lambda _{i}^{k}=m_{k}\left( A_{\mathcal{G}%
}\right) ,\end{aligned}$$for all $k\geq 0$.

In [@Las11], Lasserre proposed a technique to compute the smallest interval $\left[ a,b\right] $ containing the support[^7] of a positive Borel measure $\mu $ from its complete sequence of moments $\left( m_{r}\right) _{r\geq 0}$. In our spectral problem, the positive Borel measure under consideration is the spectral density $\mu _{\mathcal{G}}\left( x\right) $, defined in ([Spectral Measure]{}). Hence, in the context of our problem, the sequence of moments $\left( m_{r}\right) _{r\geq 0}$ is equal to $\left( m_{r}\left( A_{%
\mathcal{G}}\right) \right) _{r\geq 0}$, and the smallest 