A construction for infinite families of semisymmetric graphs revealing their full automorphism group

We give a general construction leading to different non-isomorphic families ?n,q(K) of connected q-regular semisymmetric graphs of order 2q n+1 embedded in PG(n+1,q), for a prime power q=p h , using the linear representation of a particular point set K of size q contained in a hyperplane of PG(n+1,q). We show that, when K is a normal rational curve with one point removed, the graphs ?n,q(K) are isomorphic to the graphs constructed for q=p h in Lazebnik and Viglione (J. Graph Theory 41, 249-258, 2002) and to the graphs constructed for q prime in Du et al. (Eur. J. Comb. 24, 897-902, 2003). These graphs were known to be semisymmetric but their full automorphism group was up to now unknown. For q?n+3 or q=p=n+2, n?2, we obtain their full automorphism group from our construction by showing that, for an arc K, every automorphism of ?n,q(K) is induced by a collineation of the ambient space PG(n+1,q). We also give some other examples of semisymmetric graphs ?n,q(K) for which not every automorphism is induced by a collineation of their ambient space.

Publication year:2014

Keywords:Semisymmetric graph, Linear representation, Automorphism group, Arc, Normal rational curve