On 2-distance-transitive circulant digraphs

RESUMO

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 $\K_{\frac{n}{2},\frac{n}{2}}$, the complete multipartite graph $\K_{m[b]}$ with $m\geq 3,b\geq 2$, the graph $\K_{\frac{n}{2},\frac{n}{2}}-\frac{n}{2}\K_2$ 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 $r\geq 3,b\geq 2$ and $rb=n$, the lexicographic product oriented graph \( G(p^m,r)[\overline{\K}_d]\) where \(G(p^m,r)\) obeys Condition~\ref{p-power-normal-2dt-cond}.