Semisymmtric graphs of order $2p^3$
Presented by Prof. shaofei DU
Type: Oral presentation
Track: Group Actions
A simple graph is called semisymmetric if it is regular, edge-transitive but not vertex-transitive. It is easy to see that every semisymmetric graph must be a bipartite graph with two parts of equal size, and its automorphism group acts transitively on each part. The semisymmetric graphs were first studied by Folkman in 1967, who proved that there exist no semisymmstric graphs of order $2p$ and $2p^2$, where $p$ is a prime. Since then, much work were done on such graphs. For instance, in 1985 Iofinova and Ivanov classified cubic semisymmetric graphs whose automorphism group acts primitively on both parts; and in 2000, Du and Xu classified semisymmetric graphs of order $2pq$, where $p$ and $q$ are distinct primes. Wang, Malnic and Marusic proved that the Grey graph which has order $54$ is the unique cubic semisymmtric graph of $2p^3$, where $p$ is a prime. As we know, this might be the only result about the semisymmetric graphs of order $2p^3$. Therefore, to determine the semisymmtric graphs of such order order is still one of attractive and difficult problems. In this talk, we shall show a partial classification for the semisymmtric graphs of order $2p^3$, whose automorphism group acts unfaithfully on at lest one part. This would be the first step leading to the final classification.