  The purpose of this article is to give a characterization of families of
expander graphs via right-angled Artin groups. We prove that a sequence of
simplicial graphs $\{\Gamma_i\}_{i\in\mathbb{N}}$ forms a family of expander
graphs if and only if a certain natural mini-max invariant arising from the cup
product in the cohomology rings of the groups
$\{A(\Gamma_i)\}_{i\in\mathbb{N}}$ agrees with the Cheeger constant of the
sequence of graphs, thus allowing us to characterize expander graphs via
cohomology. This result is proved in the more general framework of \emph{vector
space expanders}, a novel structure consisting of sequences of vector spaces
equipped with vector-space-valued bilinear pairings which satisfy a certain
mini-max condition. These objects can be considered to be analogues of expander
graphs in the realm of linear algebra, with a dictionary being given by the cup
product in cohomology, and in this context represent a different approach to
expanders that those developed by Lubotzky-Zelmanov and Bourgain-Yehudayoff.
