Every semisymmetric graph is necessarily bipartite, with the two parts having equal size and the automorphism
group acting transitively on each of these parts. The numbers of regular bipartite
graphs on ,
2, ... nodes are 1, 2, 1, 3, 1, 4, 1, 6, 1, ... (OEIS A087114).

Folkman (1967) proved that there are no semisymmetric graphs of order , where is a prime number and constructed
some semisymmetric graphs of order , where and are primes and , including the so-called Folkman
graph. Folkman (1967) also asked if there exists a semisymmetric graph of order
30, which was subsequently answered in the negative by Ivanov (1987).

There are no semisymmetric graphs on fewer than 20 vertices (Skiena 1990, p. 186). Examples of semisymmetric graphs are illustrated above and summarized in the following table.

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