The $Q_1$-matrix completion problem

The $Q_1$-matrix completion problem

**Authors : **

Kalyan Sinha

**Author Address : **

Department of Mathematics, PSTDS Vidyapith, Chinsurah, Hooghly, West Bengal-712305, India.

**Abstract : **

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.

**Keywords : **

Partial matrix, Matrix completion, $Q_1$-matrix, $Q_1$-completion, Digraph.

**DOI : **

**Article Info : **

*Received : * November 12, 2017; *Accepted : * February 23, 2018.