Abstract
In this paper, the nonnegative $Q$-matrix completion problem is studied. A real $n\times n$ matrix is a $Q$-matrix if for $k\in \{1,\ldots, n\}$, the sum of all $k \times k$ principal minors is positive. A digraph $D$ is said to have nonnegative $Q$-completion if every partial nonnegative $Q$-matrix specifying $D$ can be completed to a nonnegative $Q$-matrix. For nonnegative $Q$-completion problem, necessary conditions and sufficient conditions for a digraph to have nonnegative $Q$-completion are obtained. Further, the digraphs of order at most four that have nonnegative $Q$-completion have been studied.
Highlights
A partial matrix is a rectangular array of numbers in which some entries are specified while others are free to be chosen
A digraph D is said to have nonnegative Q-completion if every partial nonnegative Q-matrix specifying D can be completed to a nonnegative Q-matrix
The following example shows that a partial nonnegative Q-matrix M may have Q-completion, even when M (α) does not have
Summary
A partial matrix is a rectangular array of numbers in which some entries are specified while others are free to be chosen. A partial P0-matrix (partial P -matrix ) is one in which all fully specified principal minors are nonnegative (positive). Matrix completion problems for several classes of matrices including the classes of P and P0-matrices have been studied by a number of authors (e.g., [2, 3, 5, 7, 8, 10, 11]). Positive).) Further, the authors classified all digraphs of order up to order 4 as to Q-matrix completion. [4] Let D = Kn be an order n digraph that includes all loops and has Q-completion. We make a combinatorial study of the completion problem of partial nonnegative Q-matrices in which digraphs will play an important role
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