A classification of locally-quasiprimitive circulant digraphs

Circulant digraphs are Cayley digraphs over finite cyclic groups and constitute a fundamental class of objects in algebraic graph theory. Extending the classification of locally-primitive circulant graphs \cite{JZ-2026}, we completely determine all locally-quasiprimitive circulant digraphs. Our main theorem shows that a connected locally-quasiprimitive circulant digraph is isomorphic to one of the following: the complete graph $\K_n$, the complete bipartite graph $\K_{n/2,n/2}$, the graph $\K_{n/2,n/2}-\frac{n}{2}\K_2$ (with $n/2$ odd), the cycle $\C_n$, the directed cycle $\vec \C_n$, a normal circulant digraph of prime valency, the lexicographic product $\vec \C_m[\overline{\K_b}]$, or the tensor product $\vec \C_m\times \K_b$ with $\gcd(m,b)=1$.