双回路图的部分逆M矩阵完备

Completions for Partial Inverse M-matrices of Double Cycles

逆M矩阵是一类非常重要的非负矩阵,在生物学、物理学等很多领域中都得到了广泛的应用.利用图论理论研究逆M矩阵的完备问题是逆M矩阵研究领域的一个重要方向.双回路图是由两条简单有向回路在任意多个顶点处相交所构建的有向图.本文对此类图形的逆M矩阵完备问题进行了研究,给出了此类图形所对应的部分矩阵在两种不同情况下具有逆M矩阵完备的充分必要条件:当双回路图中的顶点均已知时,每一条回路的回路积均小于其对角元素的乘积;当双回路图中可包含未知顶点时,每一条回路至少包含一个未知顶点.同时,本文给出了具体的完备算法,并通过数值算例验证了算法的有效性.

The inverse M-matrix is a class of very important nonnegative matrices, which has been widely used in many fields such as biology and physics. Using graph theory to study the completion of inverse M-matrix is an important direction in the field of inverse M-matrices. A double cycle graph is a directed graph built from two simple directed cycles intersecting at any number of vertices in structure. In this paper, we discuss the inverse M-matrix completion problem for this class of graphs. The necessary and sufficient conditions are presented for partial matrices having inverse M-matrix completions in two different cases, whose associated graphs are double cycles. The conditions are as follows： when all vertices in the double cycle graph are specified, the cycle product of each cycle is less than the product of its diagonal elements; when the vertices in the double cycle graph include unspecified vertices, each cycle contains at least one unspecified vertex. Furthermore, we present the specific completion algorithm, and the effectiveness of the algorithms is demonstrated by a numerical example.