本原对称有向图上广义指数的极图

设D是n阶有向图(允许有环但不允许有重复弧),X C V(D),集指数expD(X)是这样的最小正整数P,使得对D中每个点v,存在从X的至少一个点到V的长为P的途径.若这样的正整数P不存在,则定义expD(X)=∞.D的第k重上广义指数F(D,k):=max{expD(X)| X C V(D),|X|=k},1≤k≤n.如果F(D,k)<∞,则称D是k-上本原的.本文完全刻划了k-上本原对称有向图的第k重上广义指数的极图.

The Extremal Digraphs of the $k$th Upper Generalized Exponent for $k$-Upper Primitive Symmetric Digraphs

Let $D$ be a digraph (loops are permitted but no multiple arcs) of order $n,$ and let $Xsubseteq V(D).$The set exponent $exp_D(X)$ is defined to be the smallest positive integer $p$ such that for each vertex $v$ of $D,$ there exists a walk of length $p$ from at least one vertex in $X$ to $v.$ If no such $p$ exists, then we define $exp_D(X)=infty .$ Let $1leq kleq n.$ Then $F(D, k):$ $=max{exp_D(X)|, Xsubset V(D),|X|=k}$ is called the $k$th upper generalized exponent of $D.$ $D$ is said to be $k$-upper primitive if $F(D,k)