Semi-symmetric graph
In the mathematical field of graph theory, a semi-symmetric graph is an undirected graph that is edge-transitive and regular, but not vertex-transitive.
In other words, a graph is semi-symmetric if each vertex has the same number of incident edges, and there is a symmetry taking any of its edges to any other of its edges, but there is some pair of vertices that cannot be mapped into each other by a symmetry. A semi-symmetric graph must be bipartite, and its automorphism group must act transitively on each of the two vertex sets of the bipartition. In the diagram on the right, green vertices can not be mapped to red ones by any automorphism.
Semi-symmetric graphs were first studied by Folkman in 1967, who discovered the smallest semi-symmetric graph, the Folkman graph on 20 vertices.[1]
The smallest cubic semi-symmetric graph is the Gray graph on 54 vertices. It was first observed to be semi-symmetric by Bouwer (1968). It was proven to be the smallest cubic semi-symmetric graph by Dragan Marušič and Aleksander Malnič.[2]
All the cubic semi-symmetric graphs on up to 768 vertices are known. According to Conder, Malnič, Marušič and Potočnik, the four smallest possible cubic semi-symmetric graph after the Gray graph are the Iofinova–Ivanov graph on 110 vertices, the Ljubljana graph on 112 vertices,[3] a graph on 120 vertices with girth 8 and the Tutte 12-cage.[4]
References
- ^ Folkman, J. (1967), "Regular line-symmetric graphs", Journal of Combinatorial Theory, 3: 215–232, doi:10.1016/S0021-9800(67)80069-3.
- ^ Bouwer, I. Z. (1968), "An edge but not vertex transitive cubic graph", Bulletin of the Canadian Mathematical Society, 11: 533–535.
- ^ Conder, M.; Malnič, A.; Marušič, D.; Pisanski, T.; Potočnik, P. (2002), "The Ljubljana Graph" (PDF), IMFM Preprints, vol. 40, no. 845, Ljubljana: Institute of Mathematics, Physics and Mechanics.
- ^ Conder, Marston; Malnič, Aleksander; Marušič, Dragan; Potočnik, Primož (2006), "A census of semisymmetric cubic graphs on up to 768 vertices", Journal of Algebraic Combinatorics, 23: 255–294, doi:10.1007/s10801-006-7397-3.