where each circle has coordinates $\left( -\lambda _{n}\left( G_{i}\right) ,-\alpha _{2}\left(
G_{i}\right) \right) $, for all $G_{i}\in \mathbf{G}$. Although $\alpha
_{2}\left( G_{i}\right) $ is a looser bound than $\beta _{2}\left(
G_{i}\right) $, we observe how there is a strong correlation between the value of $\lambda _{n}\left( G_{i}\right) $ and $\alpha _{2}\left(
G_{i}\right) $.

In these numerical experiments, we have first showed that $\alpha _{2}\left(
G_{i}\right) $ and $\beta _{2}\left( G_{i}\right) $ bound the smallest and the largest eigenvalues of the adjacency matrix, and that these bounds are tight, specially $\beta _{2}\left( G_{i}\right) $. Since these bounds can be written as explicit functions of the structural properties in $S\left(
G_{i}\right) $, we can estimate the impact of structural perturbations on the spectral radius and the smallest eigenvalue by studying $\left. \partial
\beta _{2}\right/ \partial p_{i}$ and $\left. \partial \alpha _{2}\right/
\partial p_{i}$ for $p_{i}\in S\left( G_{i}\right) $. (Details of this perturbation analysis are left for future work due to space limitations.)

![Scatter plot of $-\protect\lambda _{\min }\left( G_{i}\right) $ versus the bound $-\protect\alpha _{2}\left( G_{i}\right) $ , where each point is associated with one of the $100$ social subgraphs considered in our experiments.](BoundMin.png){width="43.00000%"}

Conclusions
===========

In this paper, we have studied 