Realizations of crystal nets. I. (Generalized) derived graphs.

A crystal net can be derived from a `generalized' voltage graph representing a graph analog of a fundamental domain of that crystal net along with a sufficient collection of its symmetries. The voltage assignments include not only isometries to the (oriented) edges, but also `weight' groups assigned to vertices for generating the vertex figures around those vertices. By varying the voltage assignments, one obtains geometrically distinct - and occasionally topologically distinct - Euclidean graphs. The focus here is on deriving simple graphs, i.e. graphs with no loops or lunes, especially uninodal edge transitive graphs.