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

2025/11/06 by Amitayu Banerjee, Banerjee, Amitayu
Mathematics · #05E18 #Combinatorics (math.CO) #FOS: Mathematics #Finite Group Theory Research #Geometric and Algebraic Topology #Group Theory (math.GR) #Limits and Structures in Graph Theory #Primary 05C25 #Secondary 20B25

paper · pdf · doi:10.48550/arxiv.2511.04151

openalex publication_date 2025/11/06 · openalex created_date 2025/11/08 · openalex updated_date 2026/07/28

Abstract

Let G be a finite group and let 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 x^-1y ∈ 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 such Cayley graphs for subsets S of size at least four. Our main results are as follows: 1. using a classical result of Burnside and Schur, we determine the automorphism groups of Cayley graphs on dihedral groups of order 2p, where p ranges over infinitely many primes and S consists only of rotations; 2. if S consists of 4 ≤ 2k < n distinct rotations, then the Cayley graph Cay(D_2n,S) is the disjoint union of two isomorphic circulant graphs on n vertices, and 3. if S is a generating set of 4≤ k≤ n reflections, then the Cayley graph Cay(D_2n,S) is bipartite, forming the disjoint union of k perfect matchings.

Citations

• Automorphism groups of a class of cubic Cayley graphs on symmetric groups
• A new look at the Burnside-Schur theorem

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• Cayley graphs on non-isomorphic groups
• Finding automorphism groups of double coset graphs and Cayley graphs are equivalent
• Topology of Cayley Graphs Applied to Inverse Additive Problems
• On The automorphism groups of us-Cayley graphs