A classification of two-distance-transitive Cayley graphs over the generalized quaternion groups

TL;DR

This work classifies all connected $2$-distance-transitive Cayley graphs over the generalized quaternion groups $Q_{4n}$ by combining normal quotient reduction with regular cyclic covers. The authors show that, after ruling out the quasiprimitive case which would force a complete graph, the action is bi-quasiprimitive or reduces to cyclic covers of a finite set of base graphs, yielding a comprehensive list of possible graphs such as $K_{x[y]}$, $K_{2n,2n}$ and several incidence/voltage-constructed families like $B(PG(d,q))$, $B'(PG(d,q))$, $X_1(4,q)$, and $\Gamma(d,q,r)$. The main theorem thus provides a complete taxonomy of $2$-distance-transitive Cayley graphs on $Q_{4n}$, expanding beyond previously known classifications for circulants and dihedrants and offering a framework for analyzing similar questions for other non-abelian groups. The results have implications for algebraic graph theory by clarifying how quotient and covering techniques interact with distance-transitivity in the Cayley setting.

Abstract

A non-complete graph is \emph{$2$-distance-transitive} if, for $i=1,2$ and for any two vertex pairs $(u_1,v_1)$ and $(u_2,v_2)$ with the same distance $i$ in the graph, there exists an element of the graph automorphism group that maps $(u_1,v_1)$ to $(u_2,v_2)$. This is a generalization concept of the classical well-known distance-transitive graphs. In this paper, we completely determine the family of $2$-distance-transitive Cayley graphs over the generalized quaternion groups.

Figures (1)

• Figure 1: ${\rm K}_{4[2]}$

Theorems & Definitions (19)

• Example 1.1
• Theorem 1.2
• Lemma 2.1
• Lemma 2.2
• Theorem 2.3
• Lemma 2.4
• Theorem 2.5
• Theorem 2.6
• Lemma 2.7
• Proposition 2.8