have expressions for the spectral moments in terms of local structural properties, our bounds relate the eigenvalues of a network with these properties. There is a large literature studying the relationship between structural and spectral properties of graphs (see [CDS80]{},[@DK04], and references therein, for an extensive list of spectral results). For many real-world networks, there is a particular set of structural properties that play a key role in the network’s functionality. For example, it is well-known that social networks contain a large number of triangles (and other cycles). Hence, it would be useful to have spectral bounds where these structural features are jointly represented. In this section, we derive new upper bounds on the smallest eigenvalue of the adjacency matrix in terms of the structural properties involved in (\[Moments as Averages\]), (\[Fourth Moment Subgraphs\]) and (\[Fifth Moment Subgraphs\]). Our results can be easily extended to derive lower bounds on the spectral radius of the adjacency matrix, although this bound is not of relevance in our analysis of games with linear best responses.

Now, we derive bounds on the smallest eigenvalue of the adjacency matrix in terms of relevant structural properties by adapting the optimization framework proposed in [@Las11]. We first 