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| THE SECOND EXPONENT SET OF PRIMITIVE DIGRAPHS |
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Citation： |
MIAO Zhengke,ZHANG Kemin.THE SECOND EXPONENT SET OF PRIMITIVE DIGRAPHS[J].Chinese Annals of Mathematics B,2000,21(2):233~236 |
| Page view： 1055
Net amount： 768 |
Authors： |
MIAO Zhengke; ZHANG Kemin |
Foundation： |
Project supported by the National Natural Science Foundation of China and the Jiangsu Provincial
Natural Science Foundation of China. |
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| Abstract： |
Let $D=(V,E)$ be a primitive digraph. The exponent of $D$, denoted by $\gamma(D)$,
is the least integer $k$ such that for any $u,v\in V$ there is a directed walk
of length $k$ from $u$ to $v$.
The local exponent of $D$ at a vertex $u\in V$, denoted by $\exp _{_D}(u)$, is the
least integer $k$ such that there is a directed walk of length $k$ from $u$ to
$v$ for each $v\in V$. Let $V=\{ 1,2,\cdots, n\} $. Following [1],
the vertices of $V$ are ordered so that $\exp _{_D}(1)\leq \exp _{_D}(2)\leq \cdots
\leq \exp _{_D}(n)=\gamma (D)$. Let $E_n(i):=\{ \exp _{_D}(i)\mid D\in PD_n\} $,
where $PD_n$ is the set of all primitive digraphs of order $n$. It is known
that $E_n(n)=\{ \gamma (D)\mid D\in PD_n\} $ has been completely settled by
[7]. In 1998, $E_n(1)$ was characterized by [5]. In this paper, the authors
describe $E_n(2)$ for all $n\geq 2$. |
Keywords： |
Primitive digraph, Local exponent, Gap |
Classification： |
05C20, 05C50, 15A33 |
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