Hamiltonicity via cohomology of right-angled Artin groups.

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.