ABSTRACT

Given disjoint non-empty subsets S and T of $\{1,\dots ,n-1\}$, a digraph D with the vertex set $\{1,2,\dots ,n\}$ is called a directed Toeplitz graph provided the arc $i\to j$ occurs if and only if $i-j\in S$ or $j-i\in T$. We investigate strong connectivity and primitivity of directed Toeplitz graphs. We prove that any primitive directed Toeplitz graph with $n\ge 6$ vertices has exponent at least 3 and for each $n\ge 6$, there is a primitive directed Toeplitz graph of order n which has exponent 3. By Wielandt's result, we know that $\exp \left(D\right)\le (n-1{)}^2+1$ for a primitive digraph D of order n. We characterize the primitive directed Toeplitz graph, for which the upper bound it attained.

摘要

给定不相交的非空子集S和T$\{1,\dots ,n-1\}$, 一个有向图D与顶点集$\{1,2,\dots ,n\}$被称为有向 Toeplitz 图，前提是弧$\mathrm{一世}\to j$当且仅当发生$\mathrm{一世}-j\in \mathrm{小号}$或者$j-\mathrm{一世}\in 吨$. 我们研究有向 Toeplitz 图的强连通性和原始性。我们证明了任何原始的有向 Toeplitz 图$n\ge 6$顶点的指数至少为 3 并且对于每个$n\ge 6$，有一个原始的n阶有向 Toeplitz 图，其指数为 3。根据 Wielandt 的结果，我们知道$\mathrm{经验}\left(D\right)\le (n-1{)}^2+1$对于n阶的原始有向图D。我们描述了原始有向 Toeplitz 图，它达到了上限。