Periodic graphs with coincident edges: folding-ladder and related graphs.

Ladder graphs admit a maximum-symmetry embedding in which edges coincide. In folding ladders, there are no zero-length edges. We give examples of high-symmetry 3-periodic ladders, particularly emphasizing the structures of 3-periodic vertex- and edge-transitive folding ladders. For these, the coincident-edge configuration is one of maximum volume for fixed edge length and has the same coordinates as (is isomeghethic to) a higher-symmetry 3-periodic graph.