A regular graph that is edge-transitive but not vertex-transitive is called a semisymmetric graph (Marušič and Potočnik 2001). In contrast, any graph that is both edge-transitive and vertex-transitive is called a symmetric graph. Note that it is possible for a graph to be edge-transitive yet neither vertex-transitive not regular. An example is the Pasch graph, which is therefore not semisymmetric. Other examples include the star graphs, rhombic dodecahedral graph, rhombic triacontahedral graph, and Schläfli double sixes graph.

cubic semisymmetric graph | edge-transitive graph | Folkman graph | Gray graph | Iofinova-Ivanov graphs | Ivanov-Ivanov-Faradjev graph | Ljubljana graph | symmetric graph | vertex-transitive graph