Author Address :
Department of Mathematics, PSTDS Vidyapith, Chinsurah, Hooghly, West Bengal-712305, India.
A matrix is a $Q_1$-matrix if it is a $Q$-matrix with positive diagonal entries. A digraph $D$ is said to have $Q_1$-completion if every partial $Q_1$-matrix specifying $D$ can be completed to a $Q_1$-matrix. In this paper, necessary and sufficient conditions for a digraph to have $Q_1$-completion are obtained. Later on the relationship among the completion problem of $Q_1$-matrix and some other class of matrices are discussed. Finally, the digraphs of order at most four that include all loops and have $Q_1$-completion are characterized.
Partial matrix, Matrix completion, $Q_1$-matrix, $Q_1$-completion, Digraph.
Article Info :
Received : November 12, 2017; Accepted : February 23, 2018.