smallest eigenvalue of the adjacency matrix determines the capacity of the network to absorb perturbations on the actions of the agents.

Motivated by the results of Bramoullé et al. [@BKA10], we study how local structural properties of the network affect the smallest eigenvalue of the adjacency matrix of interactions, affecting game equilibria. Therefore, our results build a bridge between structural properties of a network of interactions and the outcome of games with linear best responses. In particular, we use algebraic graph theory and convex optimization to derive bounds on the smallest eigenvalue of the adjacency matrix in terms of the distribution of degrees, cycles, and other important substructures. As we illustrate with numerical simulations in online social networks, these bounds can be used to estimate the effect of structural perturbations on the smallest eigenvalue.

The paper is organized as follows. In the next subsection, we review graph-theoretical terminology needed in our derivations. In Section [Strategic Section]{}, we review the relationship between the equilibria of games with linear best responses in a network and the smallest eigenvalue of the adjacency matrix of interactions. In Section \[Spectral Section\], we use algebraic graph theory to derive closed-form expressions for the so-called spectral moments 