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Automorphism Groups and Structure of 4-Valent Cayley Graphs on Dihedral Groups

Amitayu Banerjee

TL;DR

The paper advances the understanding of automorphism groups for 4-valent Cayley graphs on dihedral groups by classifying cases based on the generating set type. It leverages Burnside–Schur theory for rotation-only generating sets, shows that rotation-only Cayley graphs decompose into two identical circulants, and demonstrates bipartite structures arising from reflection-generated sets, expressed as unions of perfect matchings. For mixed configurations (two rotations and two reflections, or three rotations with one reflection, or three reflections with one rotation), it provides precise structural descriptions of the graphs and explicit automorphism-group characterizations, including wreath-product and affine-stabilizer formulations. The results extend earlier work on dihedral Cayley graphs, clarifying normality and symmetry properties in higher-valency cases with potential applications to symmetry-aware graph constructions and design.

Abstract

Let G be a finite group and S be an inverse-closed subset of G not containing the identity. The Cayley graph Cay(G, S) has vertex set G, where two vertices x and y are adjacent if and only if the group element obtained by applying the group operation to x and the inverse of y belongs to S. Kaseasbeh and Erfanian (2021) determined the structure of all Cayley graphs on the dihedral group of order 2n for subsets S of size at most three. We extend their work by analyzing the structure of all such Cayley graphs for subsets S of size four. Among other results, our main results are as follows for subsets S of size at least four: (1) using a classical result of Burnside and Schur (1911), we determine the automorphism groups of a family of Cayley graphs where S contains only rotations; (2) if S consists only of rotations, then the Cayley graph is the disjoint union of two isomorphic circulant graphs on n vertices; and (3) if S is a set of k reflections generating the dihedral group, then the Cayley graph is bipartite, forming the disjoint union of k perfect matchings.

Automorphism Groups and Structure of 4-Valent Cayley Graphs on Dihedral Groups

TL;DR

The paper advances the understanding of automorphism groups for 4-valent Cayley graphs on dihedral groups by classifying cases based on the generating set type. It leverages Burnside–Schur theory for rotation-only generating sets, shows that rotation-only Cayley graphs decompose into two identical circulants, and demonstrates bipartite structures arising from reflection-generated sets, expressed as unions of perfect matchings. For mixed configurations (two rotations and two reflections, or three rotations with one reflection, or three reflections with one rotation), it provides precise structural descriptions of the graphs and explicit automorphism-group characterizations, including wreath-product and affine-stabilizer formulations. The results extend earlier work on dihedral Cayley graphs, clarifying normality and symmetry properties in higher-valency cases with potential applications to symmetry-aware graph constructions and design.

Abstract

Let G be a finite group and S be an inverse-closed subset of G not containing the identity. The Cayley graph Cay(G, S) has vertex set G, where two vertices x and y are adjacent if and only if the group element obtained by applying the group operation to x and the inverse of y belongs to S. Kaseasbeh and Erfanian (2021) determined the structure of all Cayley graphs on the dihedral group of order 2n for subsets S of size at most three. We extend their work by analyzing the structure of all such Cayley graphs for subsets S of size four. Among other results, our main results are as follows for subsets S of size at least four: (1) using a classical result of Burnside and Schur (1911), we determine the automorphism groups of a family of Cayley graphs where S contains only rotations; (2) if S consists only of rotations, then the Cayley graph is the disjoint union of two isomorphic circulant graphs on n vertices; and (3) if S is a set of k reflections generating the dihedral group, then the Cayley graph is bipartite, forming the disjoint union of k perfect matchings.

Paper Structure

This paper contains 11 sections, 15 theorems, 5 equations, 1 figure, 2 tables.

Table of Contents

  1. Introduction
  2. Results
  3. Preliminaries
  4. Only rotations
  5. Automorphism groups
  6. Four reflections
  7. Automorphism groups
  8. Two rotations and two reflections
  9. Automorphism groups
  10. One rotation or one reflection
  11. Appendix A

Key Result

Corollary 2.10

(of Fact Fact 2.9(2)) Let $p$ be a prime. Let $G\leq S_p$ be a transitive permutation group of degree $p$ that contains a regular cyclic subgroup. Then $G$ is primitive and $G$ is either

Figures (1)

  • Figure 1: The graph $\mathrm{Cay}(D_{12}, \{r^{\pm1}, r^{\pm2}\})$ can be expressed as $K_{2,2,2} + K_{2,2,2}$.

Theorems & Definitions (66)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • Definition 2.7
  • Definition 2.8
  • Corollary 2.10
  • Proposition 3.1
  • ...and 56 more