主对角元全为零的Z矩阵的组合性质

Combinatorial properties of Z-matrices with zero diagonal

Fiedler和Markham定义了n阶Lt 矩阵 ,并将所有n阶Z矩阵的集合分成n+1类 :L0 ,L1 ,… ,Ln,本文从矩阵的伴随有向图出发 ,着重研究了主对角元全为 0的Z矩阵的一些有趣的性质。首先得到一个重要定理 :主对角元全为 0的Z矩阵A属于类Lt 的充要条件是A的伴随有向图的最小圈长为t+1 ,然后利用它给出了主对角元全为 0的Lt矩阵的零位模式及其伴随有向图的刻划。

Fiedler and Markham introduced the L t -Matrices,and paritioned the set Z n of all n×n Z -matrices into n+1 classes: L 0,L 1,...,L n .In this paper we use the associated digraph of a Z-matrix to investigates the combinatorial properties of Z-matrices whose diagonal entries are all zeros(with zero diagonal).We first proove that a Z-matrix A with zero diagonal is a L t -matrix if and only if the least length of the circuits of the associated digraph of A is t+1 .Then we decribe the zero-pattern and digraph of an L t -matrix with zero diagonal.