the support of the eigenvalue spectrum to study the role of structural properties in the existence and the stability of a Nash equilibrium. For this purpose, we analyze real data from a regional network of Facebook that spans $63,731$ users (nodes) connected by $817,090$ friendships (edges) [@VMCG09]. In order to corroborate our results in different network topologies, we extract multiple medium-size social subgraphs from the Facebook graph by running a Breath-First Search (BFS) around different starting nodes. Each BFS induces a social subgraph spanning all nodes 2 hops away from a starting node, as well as the edges connecting them. We use this approach to generate a set $\mathbf{G}=\{G_{i}\}_{i\leq
100}$ of 100 different social subgraphs centered around 100 randomly chosen nodes.[^8]

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

From Corollary \[One corollary\] and Lemma \[Lemma 4th metrics\] we can compute the first five spectral moments of a graph $G_{i}$ from the following structural properties: number of nodes ($n_{i}$), edges ($e_{i}$), triangles ($\Delta _{i}$), quadrangles ($Q_{i}$), pentagons ($\Pi _{i}$), as well as the 