On 2-distance-transitive circulant digraphs

Circulant digraphs form a prominent class of Cayley digraphs defined on finite cyclic groups. Building on the existing classification of $2$-arc-transitive circulant graphs, this paper presents a complete classification of $2$-distance-transitive circulant digraphs. Our main theorem establishes that every connected $2$-distance-transitive circulant digraph is isomorphic to one of the following: the undirected cycle $C_n$, the complete bipartite graph ${\text{K}}_{\frac{n}{2},\frac{n}{2}}$, the complete multipartite graph ${\text{K}}_{m[b]}$ with $m\ge 3,b\ge 2$, the graph ${\text{K}}_{\frac{n}{2},\frac{n}{2}}-\frac{n}{2}{\text{K}}_2$ for odd $\frac{n}{2}$, prime-order Paley graphs, the directed cycle ${\overrightarrow{C}}_n$, the oriented graph $G(p^m,r)$ satisfying Condition~\ref{p-power-normal-2dt-cond}, the oriented graph $C_r(b,1)$ with $r\ge 3,b\ge 2$ and $rb=n$, the lexicographic product oriented graph $G(p^m,r)[{\bar{\text{K}}}_d]$ where $G(p^m,r)$ obeys Condition~\ref{p-power-normal-2dt-cond}.