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

We give a general construction leading to different non-isomorphic families $\varGamma_{n,q}(\mathcal{K})$ of connected q-regular semisymmetric graphs of order 2q ⁿ⁺¹ embedded in $\operatorname{PG}(n+1,q)$ , for a prime power q=p ^h , using the linear representation of a particular point set $\mathcal{K}$ of size q contained in a hyperplane of $\operatorname{PG}(n+1,q)$ . We show that, when $\mathcal{K}$ is a normal rational curve with one point removed, the graphs $\varGamma_{n,q}(\mathcal{K})$ 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 $\mathcal{K}$ , every automorphism of $\varGamma_{n,q}(\mathcal{K})$ is induced by a collineation of the ambient space $\operatorname{PG}(n+1,q)$ . We also give some other examples of semisymmetric graphs $\varGamma _{n,q}(\mathcal{K})$ for which not every automorphism is induced by a collineation of their ambient space.