对称本原有向图广义重上指数的极图刻划

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 应的无向图连通且至少包含一个奇圈。文［２］给出了具有最小奇圈长ｒ的ｎ阶对称本 原有向图广义ｋ重上指数的最大数．本文将在此基础上，给出其极图的完全刻划．

The Carcterization for the Extreme Digraphs of the k~(th) Upper Generalized Exponents

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 is connected and contains at least one odd cycle. In [2], we have determined the largest value of the k~(th) upper generalized exponents over the set of primitive symmetric digraphs whose shortest odd cycle length is a fixed number T. In this paper, we give a complete characterization for the extremal digraphs.