sum-of-squares of the degrees ($W_{i}$), and the degree-triangle correlation ($\mathcal{C}_{i}$). For convenience, we define $S\left(
G_{i}\right) \triangleq \left\{ n_{i},e_{i},\Delta _{i},Q_{i},\Pi _{i},W_{i},%
\mathcal{C}_{i}\right\} $ as a set of relevant structural properties of $%
G_{i}$. In our numerical experiment, we first measure the set of relevant properties $S_{i}$ for each social subgraph $G_{i}\in \mathbf{G}$, and then compute the first five spectral moments of its adjacency matrix. From these moments, we then compute the bounds $\alpha _{2}\left( G_{i}\right) $ and $%
\beta _{2}\left( G_{i}\right) $ using Proposition \[Lasserre bound\]. As we mentioned before, these bounds can be computed as the maximum and minimum roots of a third order polynomial, for which closed form expressions are known.

We illustrate the quality of our bounds in the following figures. Fig. 2 is a scatter plot where each circle has coordinates $\left( \rho \left(
G_{i}\right) ,\beta _{2}\left( G_{i}\right) \right) $, for all $G_{i}\in 
\mathbf{G}$. Observe how *the spectral radii* $\rho \left( G_{i}\right) 
$ *of these social subgraphs are remarkably close to the theoretical lower bound* $\beta _{2}\left( G_{i}\right) $. Therefore, we can use $\beta
_{2}\left( G_{i}\right) $ as an estimate of $\rho \left( G_{i}\right) $ for social subgraphs. In Fig. 3 we include a scatter plot 