  Let $\Gamma$ be a finite graph and let $A(\Gamma)$ be the corresponding
right-angled Artin group. We characterize the Hamiltonicity of $\Gamma$ via the
structure of the cohomology algebra of $A(\Gamma)$. In doing so, we define and
develop a new canonical graph associated to a matrix, and in doing so provide a
novel perspective on the matrix determinant. We give some applications to
complexity theory, zero--knowledge proof protocols, and models for random
graphs.
