关于2-距离传递循环有向图
On 2-distance-transitive circulant digraphs
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- Xiangtan University（湘潭大学）
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中文总结 AI 辅助
本文基于2-弧传递循环图的分类，完整分类了2-距离传递循环有向图，列出了所有同构类型，包括无向圈、完全多部图、Paley图及若干有向图族。
中文摘要 AI 辅助
循环有向图构成有限循环群上定义的一类重要的Cayley有向图。基于已有的2-弧传递循环图的分类，本文给出了2-距离传递循环有向图的完整分类。我们的主要定理表明，每个连通的2-距离传递循环有向图同构于以下之一：无向圈$C_n$，完全二部图$\K_{\frac{n}{2},\frac{n}{2}}$，完全多部图$\K_{m[b]}$（其中$m\geq 3,b\geq 2$），当$\frac{n}{2}$为奇数时的图$\K_{\frac{n}{2},\frac{n}{2}}-\frac{n}{2}\K_2$，素数阶Paley图，有向圈$\overrightarrow{C}_n$，满足条件~\ref{p-power-normal-2dt-cond}的有向图$G(p^m,r)$，满足$r\geq 3,b\geq 2$且$rb=n$的有向图$C_r(b,1)$，以及字典序积有向图$G(p^m,r)[\overline{\K}_d]$，其中$G(p^m,r)$满足条件~\ref{p-power-normal-2dt-cond}。
英文摘要
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}.
