Semi-symmetric graph

Semi-symmetric graph

In mathematics, 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.

Semi-symmetric graphs were first studied by harvtxt|Folkman|1967, who discovered the smallest semi-symmetric graph, the Folkman graph on 20 vertices. The smallest cubic semi-symmetric graph is the Gray graph on 54 vertices. [The Gray graph was first observed to be semi-symmetric by harvtxt|Bouwer|1968. It was proven to be the smallest cubic semi-symmetric graph by Dragan Marušič and Aleksander Malnič.] A complete list of cubic semi-symmetric graphs on at most 768 vertices is given by harvtxt|Conder|Malnič|Marušič|Potočnik|2006.

Notes

References

*citation
last = Bouwer | first = I. Z.
title = An edge but not vertex transitive cubic graph
journal = Bulletin of the Canadian Mathematical Society
volume = 11 | pages = 533–535 | year = 1968.

*citation
last1 = Conder | first1 = Marston
last2 = Malnič | first2 = Aleksander
last3 = Marušič | first3 = Dragan | authorlink3 = Dragan Marušič
last4 = Potočnik | first4 = Primož
title = A census of semisymmetric cubic graphs on up to 768 vertices
journal = Journal of Algebraic Combinatorics
volume = 23 | year = 2006 | pages = 255–294
doi = 10.1007/s10801-006-7397-3.

*citation
first = J. | last = Folkman
title = Regular line-symmetric graphs
journal = Journal of Combinatorial Theory
volume =3 | year = 1967 | pages = 215–232.

External links

*mathworld | urlname = SemisymmetricGraph | title = Semisymmetric Graph


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