对称有向图的广义本原指数集

一个有向图Ｄ称为本原有向图，若存在某自然数ｋ，使Ｄ中任一点ｕ到任一点ｖ都有长为ｋ之途径。若Ｄ是一个对称有向图，则Ｄ是本原的当且仅当Ｄ对应的无向图Ｇ连通且至少包含一个奇圈。本文研究最小奇圈长为ｒ的ｎ阶对称本原有向图，完全刻划了第一类广义本原指数集，并部分地解决了第三类广义本原指数集的刻划问题。

THE SET OF GENERALIZED EXPONENTS OF SYMMETRIC PRIMITIVE DIGRAPHS

A digraph D of order n is called primitive if there exists a positive integer k such that for each ordered pair of vertices u and v, there is a walk of length k from u to v. If D is a symmetric digraph, then D is primitive if and only if its corresponding graph G is connected and contains at least one odd cycle. In this paper, we research primitive symmetric digraph of order n whose shortest odd cycle length is a fixed number r. We characterized generalized exponent set completely and characterized the kth upper generalized exponent set in parts.