对称本原图的集指数与本原简单图的广义上指数的极图

一个有向图D称为本原的，如果存在某个正整数k，使得对于D中的任一点x到任一点y都有长为k的途径，这样的正整数k中的最小者称为D的本原指数，作为本原指数概念的推广，R．A．Brualdi和柳柏濂于1990年引入了本原有向图的广义本原指数的新概念，本文给出了对称本原图的集指数的一些性质，并对本原简单图的广义上指数的极图进行了完全刻划。

THE SET EXPONENT OF SYMMETRIC PRIMITIVE DIGRAPHS AND THE EXTREMAL GRAPHS OF THE KTH UPPER GENERALIZED EXPONENT FOR PRIMITIVE SIMPLE GRAPHS

A directed graph D is called primitive if there exists a positive integer k such that for each ordered pair of vertices x and y (not necessarily distinct), there is a walk of length k from x to y. The smallest such k is called the exponent of D. As a generalization of exponent, R. A. Brualdi and Bolian Liu introduced the new concept of generalized exponent for primitive digraphs in 1990. In this paper, we give some qualities of the set exponent of symmetric primitive digraphs, and characterize completely the extremal graphs of the kth upper generalized exponent for primitive simple graphs.