无环的本原反对称带号有向图的局部基与基指数

THE LOCAL BASES AND BASES OF PRIMITIVE ANTI-SYMMETRIC SIGNED DIGRAPHS WITH NO LOOPS

研究了n阶无环的本原反对称带号有向图S的局部基lS(k),得到了lS(k)≤max{n+l-1,n+k-1}(l为S中最小奇圈的长),给出了k≥l时lS(k)=n+k-1的一个极图,因此证明了n阶无环的本原反对称带号有向图S的基指数l(S)≤2n-1,给出了达到上界的极图.

Let S be a signed digraph,if the underlying digraph D（S） is symmetric,and each 2-cycle in S is negative,then S is called an anti-symmetric signed digraph.The local bases of primitive anti-symmetric signed digraphs with no loops of order n is studied, and the following conclusion is proved that lS（k）≤ max {n＋l-1,n＋k-1},where l is the shortest length of odd cycles of S. The upper bounds of the bases of primitive anti-symmetric signed digraphs with no loops of order n are also obtained, and it is shown that the above obtained upper bounds are sharp.