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

Cara, Philippe ; Rottey, Sara ; Van de Voorde, Geertrui

Journal of Algebraic Combinatorics, Tome 39 (2014) no. 4, pp. 967-988.

Résumé

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 [F. Lazebnik and R. Viglione [J. Graph Theory 41, No. 4, 249--258 (2002; Zbl 1012.05083)] and to the graphs constructed for $q$ prime in [S. Du et al., Eur. J. Comb. 24, No. 7, 897--902 (2003; Zbl 1026.05056)]. These graphs were known to be semisymmetric but their full automorphism group was up to now unknown. For $q\geq n+3$ or $q=p=n+2$, $n\geq 2$, we obtain their full automorphism group from our construction by showing that, for an arc $\mathcal{K}$, 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.

Détail

Classification : 05C62, 05C60, 05C75
Keywords: semisymmetric graph, linear representation, automorphism group, arc, normal rational curve