Two-distance transitive normal Cayley graphs

TL;DR

This paper constructs an infinite family of normal Cayley graphs that are 2-distance-transitive but not distance- or 2-arc-transitive, addressing a previously open question and correcting earlier literature claims.

Contribution

It introduces a new infinite family of normal Cayley graphs with specific transitivity properties, resolving an open problem and correcting prior misconceptions.

Findings

Constructed an infinite family of such graphs.

Demonstrated these graphs are 2-distance-transitive.

Showed these graphs are not distance- or 2-arc-transitive.

Abstract

In this paper, we construct an infinite family of normal Cayley graphs, which are $2$-distance-transitive but neither distance-transitive nor $2$-arc-transitive. This answers a question raised by Chen, Jin and Li in 2019 and corrects a claim in a literature given by Pan, Huang and Liu in 2015.