The Judgment Of Inverse M-Matrix Completion Based On Digraph And Its Algorithm Design & Realization

M-matrices are a class of matrices with nonpositive off-diagonal entries and nonnegative diagonal entries. And inverse M-matrices are a class of non- negative matrices whose inverse are M-matrices. Inverse M-matrices have the widespread application in many fields. In this paper, we study the inverse M-matrix completion problems making use of graphic theory. According to the different characteristics of graphs corresponding to partial matrices, we specifically discuss the inverse M-matrices completions of partial matrices associated to some special graphs, and find their respective completion methods.Firstly, we study the inverse M-matrices completion problems of path n -chordal digraph (directed graph). Based on cyclic n-chordal digraph, we define the path n -chordal directed graph ( n can be any natural number.). Then, based on the completion theorems of cyclic n-chordal digraph, we expand the simple cycle to simple path in the digraph. In this paper, the completion of simple digraph path, path 1-chordal digraph, path 2-chordal digraph and path 3-chordal digraph are studied, And the completion of path n -chordal digraph are discussed. At the same time, the completion conditions and completion algorithmic are also given respectively. Through the completion algorithmics, we can easily find the completion of the partial inverse M-matrices which graph corresponding to the digraph.Secondly, we discuss the completions of several special inverse M-matrix patterns, including the loop path pattern and cyclic n-chordal digraph, and give the corresponding completion theorems and specific completion algorithms.Finally, we use the programming language i.e. java and matlab to realize the completion algorithms of simple directed path, path 1-chordal digraph andpath 2-chordal digraph put forward in the paper. At the same time, we prove that the algorithms are feasibility and validity through the complexity analysis of the algorithms.In this paper, we combine matrix theory, graph theory and algorithm design together to study the inverse M-matrix completion problems with some special digraphs. And some results are got. The paper will play reference function for deeply research on inverse M-matrices.