A semisymmetric graph is a regular graph that is edge-transitive but not vertex-transitive (Marušič and Potočnik 2001).
Semisymmetric graphs should not be confused with half-arc-transitive graphs. Both classes are regular and edge-transitive, but not arc-transitive. However, half-arc-transitive graphs are vertex-transitive, while semisymmetric graphs are not (Conder and Žitnik 2016).
Note that it is possible for a graph to be edge-transitive yet neither vertex-transitive nor regular.
An example is the Pasch graph, which is therefore
not semisymmetric. Other examples include the star graphs 
for 
, rhombic dodecahedral
graph, rhombic triacontahedral graph,
and Schläfli double sixes graph.
A useful obstruction is that a connected graph which is edge-transitive and has an edge whose endpoints belong to the same group orbit under its automorphism group must be vertex-transitive. Every edge is an image of that edge, so both endpoints of every edge lie in the same group orbit. Since the graph is connected, all vertices belong to this group orbit. This excludes generalized Petersen graphs from being semisymmetric, since their graph automorphism rotating both rims puts adjacent outer vertices in the same group orbit.
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 graph order 
,
where 
is a prime number and constructed some semisymmetric
graphs of graph order 
, where 
and 
are primes and 
, including the so-called Folkman
graph. Folkman (1967) also asked if there exists a semisymmetric graph of graph 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.
 | graph |
| 20 | Folkman graph |
| 54 | Gray graph |
| 64 | (16,4)-Hadamard graph |
| 80 | (4,8)-cage graph |
| 110 | 110-Iofinova-Ivanov graph |
| 112 | Ljubljana graph |
| 126 | Tutte 12-cage |
| 182 | 182-Iofinova-Ivanov graph |
| 288 | Leonard graph |
| 312 | (6,8)-cage graph |
| 506 | 506-Iofinova-Ivanov graph |
| 800 | (8,8)-cage graph |
| 1640 | (10,8)-cage graph |
| 2730 | (5,12)-cage graph |
| 2928 | (12,8)-cage graph |
| 4760 | (14,8)-cage graph |
." Comm. Alg. 28, 2685-2715, 2000.Folkman,
J. "Regular Line-Symmetric Graphs." J. Combin. Th. 3, 215-232,
1967.House of Graphs. Semisymmetric Graphs.