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 ${\text{K}}_n$, the complete bipartite graph ${\text{K}}_{n\mathrm{/}2,n\mathrm{/}2}$, the graph ${\text{K}}_{n\mathrm{/}2,n\mathrm{/}2}-\frac{n}{2}{\text{K}}_2$ (with $n\mathrm{/}2$ odd), the cycle ${\text{C}}_n$, the directed cycle ${\vec{\text{C}}}_n$, a normal circulant digraph of prime valency, the lexicographic product ${\vec{\text{C}}}_m[\overline{{\text{K}}_b}]$, or the tensor product ${\vec{\text{C}}}_m\times {\text{K}}_b$ with $\gcd (m,b)=1$.