Although Chemical Reaction Networks (CRNs) form a rich class of nonlinear systems that can exhibit wide range of nonlinear behaviours, many common examples are observed to be asymptotically stable regardless of the kinetics. This paper presents the recently uncovered class of Graphically Stable Networks (GSNs) which is characterized by the existence of a robust Lyapunov function defined in the reaction coordinates. Subject to mild conditions, the existence of these functions guarantees asymptotic stability of a network regardless of the specific form of kinetics. Construction methods for these functions are provided and illustrated by examples.