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

TL;DR

This paper introduces a general construction method for infinite families of semisymmetric graphs, determines their full automorphism groups in many cases, and explores cases where automorphisms are not induced by ambient space collineations.

Contribution

The paper provides a new construction for semisymmetric graphs and characterizes their automorphism groups, extending previous results and identifying cases with non-collineation automorphisms.

Findings

Constructed infinite families of semisymmetric graphs with specific properties.

Determined the full automorphism groups for these graphs in many cases.

Identified examples where automorphisms are not induced by ambient space collineations.

Abstract

We give a general construction leading to different non-isomorphic families $\Gamma_{n,q}(\K)$ 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 $\Gamma_{n,q}(\K)$ 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 $q\geq n+3$ or $q=p=n+2$, $n\geq 2$, we obtain their full automorphism group from our construction by showing that, for an arc $\K$, every automorphism of $\Gamma_{n,q}(\K)$ is induced by a collineation of the ambient space $\PG(n+1,q)$. We also give some other examples of semisymmetric graphs $\Gamma_{n,q}(\K)$ for which not every automorphism is induced by a collineation of their ambient space.