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Symmetries of regular $q$-graphs

Daniel R Hawtin, Padraig Ó Catháin

TL;DR

This work provides a complete classification of $k$-regular q-graphs that are flag-transitive or symmetric, situating the classification within the landscape of transitive linear groups and finite geometry. The authors demonstrate that the eligible q-graphs arise from well-known geometric constructs, including Desarguesian and non-Desarguesian spreads, symplectic polar spaces, and symplectic generalized hexagons, with extensions to one-dimensional semilinear actions and Lie-type groups over extension fields. The results show that, beyond a few low-dimensional or exceptional cases, the symmetry constraints force the q-graphs to be built from classical finite-geometric objects, and that their classical counterparts are often symmetric graphs. Overall, the paper connects q-graph symmetries to the classification of transitive linear groups and, ultimately, to the finite simple groups, offering a structured map of highly symmetric q-graph examples and their origins.

Abstract

Given a finite vector space $V=\mathbb{F}_q^n$, the $q$-analogue of a graph, called a $q$-graph, is a pair $Γ=(\mathcal{V},\mathcal{E})$, where $\mathcal{V}$ is the set of $1$-dimensional subspaces of $V$ and $\mathcal{E}$ is a subset of the $2$-dimensional subspaces of $V$. Elements of $\mathcal{V}$ and $\mathcal{E}$ are called vertices and edges, respectively. If the edges through a vertex $X$ consist of all $2$-spaces of a $(k+1)$-dimensional space which contain $X$, regardless of the choice of vertex, then $Γ$ is $k$-regular. Moreover, $Γ$ is flag-transitive if there is a subgroup of $Γ{\rm L}_n(q)$ preserving $\mathcal{E}$ and acting transitively on the set of all incident vertex-edge pairs; and symmetric if there is a subgroup of $Γ{\rm L}_n(q)$ preserving $\mathcal{E}$ and acting transitively on the set of all ordered pairs of adjacent vertices. This paper classifies all $k$-regular $q$-graphs that are either flag-transitive or symmetric. The $q$-graphs in the classification are constructed from familiar objects in finite geometry, including spreads, symplectic polar spaces, and generalised hexagons. The classification depends essentially on the classification of transitive linear groups, and thus ultimately on the classification of finite simple groups.

Symmetries of regular $q$-graphs

TL;DR

This work provides a complete classification of -regular q-graphs that are flag-transitive or symmetric, situating the classification within the landscape of transitive linear groups and finite geometry. The authors demonstrate that the eligible q-graphs arise from well-known geometric constructs, including Desarguesian and non-Desarguesian spreads, symplectic polar spaces, and symplectic generalized hexagons, with extensions to one-dimensional semilinear actions and Lie-type groups over extension fields. The results show that, beyond a few low-dimensional or exceptional cases, the symmetry constraints force the q-graphs to be built from classical finite-geometric objects, and that their classical counterparts are often symmetric graphs. Overall, the paper connects q-graph symmetries to the classification of transitive linear groups and, ultimately, to the finite simple groups, offering a structured map of highly symmetric q-graph examples and their origins.

Abstract

Given a finite vector space , the -analogue of a graph, called a -graph, is a pair , where is the set of -dimensional subspaces of and is a subset of the -dimensional subspaces of . Elements of and are called vertices and edges, respectively. If the edges through a vertex consist of all -spaces of a -dimensional space which contain , regardless of the choice of vertex, then is -regular. Moreover, is flag-transitive if there is a subgroup of preserving and acting transitively on the set of all incident vertex-edge pairs; and symmetric if there is a subgroup of preserving and acting transitively on the set of all ordered pairs of adjacent vertices. This paper classifies all -regular -graphs that are either flag-transitive or symmetric. The -graphs in the classification are constructed from familiar objects in finite geometry, including spreads, symplectic polar spaces, and generalised hexagons. The classification depends essentially on the classification of transitive linear groups, and thus ultimately on the classification of finite simple groups.
Paper Structure (17 sections, 15 theorems, 10 equations)

This paper contains 17 sections, 15 theorems, 10 equations.

Table of Contents

  1. Introduction
  2. Groups acting on classical graphs
  3. Connection to classical graphs and geometries
  4. Outline of the paper
  5. Preliminaries
  6. Flag-transitive q-graphs of dimension at most four
  7. Transitive linear groups
  8. Examples
  9. Flag-transitive q-graphs
  10. Further examples of q-graphs
  11. One dimensional semi-linear case
  12. Groups of Lie type
  13. Extension fields
  14. Special linear groups over extension fields
  15. Symplectic groups over extension fields
  16. ...and 2 more sections

Key Result

Theorem 1.2

Suppose that $\Gamma=(\mathcal{V},\mathcal{E})$ is a $k$-regular and $G$-flag-transitive $q$-graph on $V=\mathbb F_q^n$ that is neither complete nor empty. Then one of the following occurs.

Theorems & Definitions (39)

  • Definition 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Corollary 1.4
  • Proposition 1.5
  • Example 1.6
  • Proposition 1.7
  • Definition 2.1
  • Lemma 2.2
  • proof
  • ...and 29 more