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

We give a general construction leading to different non-isomorphic families ${\mathrm{\Gamma }}_{n,q}(\text{K})$ of connected $q$-regular semisymmetric graphs of order $2q^{n+1}$ embedded in $\text{PG}(n+1,q)$, for a prime power $q=p^h$, using the linear representation of a particular point set $\text{K}$ of size $q$ contained in a hyperplane of $\text{PG}(n+1,q)$. We show that, when $\text{K}$ is a normal rational curve with one point removed, the graphs ${\mathrm{\Gamma }}_{n,q}(\text{K})$ are isomorphic to the graphs constructed for $q$ prime in [9] and to the graphs constructed for $q=p^h$ in [20]. These graphs were known to be semisymmetric but their full automorphism group was up to now unknown. For $q\ge n+3$ or $q=p=n+2$, $n\geq 2$, we obtain their full automorphism group from our construction by showing that, for an arc $\text{K}$, every automorphism of ${\mathrm{\Gamma }}_{n,q}(\text{K})$ is induced by a collineation of the ambient space $\text{PG}(n+1,q)$. We also give some other examples of semisymmetric graphs ${\mathrm{\Gamma }}_{n,q}(\text{K})$ for which not every automorphism is induced by a collineation of their ambient space.