  For networks of coupled dynamical systems we characterize admissible
functions, that is, functions whose gradient is an admissible vector field. The
schematic representation of a gradient network dynamical system is of an
undirected cell graph, and we use tools from graph theory to deduce the general
form of such functions, relating it to the topological structure of the graph
defining the network. The coupling of pairs of dynamical systems cells is
represented by edges of the graph, and from spectral graph theory we detect the
existence and nature of equilibria of the gradient system from the critical
points of the coupling function. In particular, we study fully synchronous and
2-state patterns of equilibria on regular graphs.These are two special types of
equilibrium configurations for gradient networks. We also investigate
equilibrium configurations of S1-invariant admissible functions on a ring of
cells.
