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\section{The $Q$-matrix completion problem} By {\sl Luz DeAlba, Leslie Hogben, Bhaba Sharma}.  

\medskip  \noindent A partial matrix is a matrix that contains some specified entries, while all other entries remain unspecified and can be freely assingned a value. An $n \times n$ partial matrix, $B$, specifies a digraph $D = \left(V_D, A_D \right)$, if $V_D = \{ 1, 2, \dots, n \}$, and $(i,j) \in A_D$ if and only if the entry $b_{ij}$ of $B$ is specified. A real $n \times n$ matrix is a $Q$-matrix if for every $k = 1, 2, \dots, n$, the sum of all $k \times k$ principal minors is positive. A partial matrix is a partial $Q$-matrix if the sum of all $k \times k$ principal minors is positive for every $k$ for which all $k \times k$ principal matrices are fully specified. A digraph $D$ is said to have $Q$-completion if every partial $Q$-matrix specifying $D$ can be completed to a $Q$-matrix. In this presentation we give sufficient conditions for a digraph to have $Q$-completion, we also give necessary conditions for a digraph to have $Q$-completion, and characterize those digraphs of order at most four that have $Q$-completion.
%Partial Matrix, matrix completion, $Q$-matrix, digraph","15A48","05C50","","12:24:31","Sat Mar 22 2008","63.224.179.146"
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