For a network of coupled dynamical systems I present the characterization of ad- missible functions, namely the functions on the total space whose gradient is an admissible vector field. The schematic representation of a network dynamical system in this context is of an undirected cell graph. I intend to discuss the nature of certain types of critical points of admissible functions related to the architecture of regular graphs. In particular, we explore critical points that correspond to fully synchronous and 2-state patterns of equilibria of the gradient vector field on these graphs. I wish also to present an analysis of all the critical points of S1-invariant admissible functions on a ring of cells. This is a joint work with Mark Roberts.