THE $Q_{N^{1}_{0}}$-MATRIX COMPLETION PROBLEM FOR DIGRAPHS 5x5 MATRICES

Abstract

An $n\times n$ matrix is called a $Q_{N^{1}_{0}}$ matrix if for every $k=\lbrace 1,2,3,\ldots ,n \rbrace $ the sum of all $ k \times k $ principal minors is non-positive. A digraph $D$ is said to have $Q_{N^{1}_{0}}$-completion if every partial $Q_{N^{1}_{0}}$-matrix specifying $D$ can be completed to a $Q_{N^{1}_{0}}$-matrix. In this paper $Q_{N^{1}_{0}}$-matrix completion problem is considered. It is shown that partial non-positive $Q_{N^{1}_{0}}$ -matrices representing all digraphs of order $5$ with $q = 0$ to $7$ have $Q_{N^{1}_{0}}$ completion.