The non-negative $Q_1$-matrix completion problem

Abstract

A matrix is a Q 1-matrix if it is a Q-matrix with positive diagonal entries. A matrix is a nonnegative matrix if it is a matrix with nonnegative entries. A digraph D is said to have nonnegative Q 1-completion if every partial nonnegative Q 1-matrix specifying D can be completed to a nonnegative Q 1-matrix. In this paper, some necessary and sufficient conditions for a digraph to have nonnegative Q 1-completion are provided. Later on the relationship among the completion problems of nonnegative Q 1-matrix and some other class of matrices are shown. Finally, the digraphs of order at most four that include all loops and have nonnegative Q 1-completion are singled out.