迹为零的几类特殊本原矩阵的本原指数

The Primitive Exponent of Several Specific Primitive Matrices with Trace Zero

对本原矩阵指数集研究中的另一方面是研究一些特殊的本原矩阵类的本原指数.邵嘉裕先生和李乔先生在这一领域取得了一些令人满意的结果[1].邵嘉裕先生[2]给出了一个特殊的本原矩阵——对称本原矩阵类的指数集合En={m∈Z+|存在某个n阶对称本原阵A,使γ(A)=m},并且给出了En的完全刻划.我们考虑一个特殊的本原矩阵类:对角元为零的几类特殊本原矩阵类的指数集.记对角元为零的本原矩阵集为T0n.证明一类对角线为零的最小圈长n-d+1的特殊本原有向图的指数集.这里的d是满足:大于等于2但小于n/2的偶数,且gcd(n,n-d+1)=1.

Another aspect in the study of the exponent set of primitive matrices focuses on the exponent set of primitive matrices of some specific primitive matrices. Chinese scholars have contributed a lot in this area, among whom Professor Shao Jiayu and Professor Li Qiao are two distinguished figures. They have achieved satisfactory discoveries. Professor Shao Jiayu proposed a special primitive matrices-the symmetric exponent set of primitive matrices and pointed out that En^～ = { m ∈ Z^＋ ｜ has an order symmetric primitive matrix A which brings the result： γ（A） = m }. He also gave a sound description of En^～ . In this thesis, we discuss one special type of primitive matrices： We discuss the several exponent set of primitive matrices on n vertices with trace zero and primitive matrices Tn^0. with trace zero The thesis proves one exponent set of primitive directed graph with trace zero and the minimum length of the circle equals n - d ＋ 1 . Here d is even which exceeds or equals 2, and less than [n/2] and ged（n,n-d＋1）=1.