Symmetric vertex- and edge-transitive 3-periodic nets

Vertex- and edge-transitive graphs that admit tilings (tessellate graphs) have been known for 20 years. In this paper, we identify additional families of vertex- and edge-transitive graphs with symmetric embeddings, for which the order of the point symmetry at a vertex equals the vertex's coordination number. Remarkably, symmetric tessellate graphs appear over 150 000 times in crystal structures, while there is only one reported occurrence of a symmetric decussate (untessellable) graph.