Completion of Weakly Sign Symmetric PO-Matrix Problem for 5 × 5 Matrices Specifying Digraphs of Order 5 with UP to 5 Arcs
Joseph Marro *
Department of Mathematics, Meru University of Science and Technology, P.O Box 972, Meru, Kenya.
Josephine Mutembei
Department of Mathematics, Meru University of Science and Technology, P.O Box 972, Meru, Kenya.
Loyford Njagi
Department of Mathematics, Meru University of Science and Technology, P.O Box 972, Meru, Kenya.
*Author to whom correspondence should be addressed.
Abstract
An n × n matrix is a weakly sign symmetric matrix if the off-diagonal elements have the property that if the entry in row i and column j is non-zero, then the entry in row j and column i must have same sign or zero. A digraph D has a Wss Po -matrix completion if every partial weakly sign symmetric Po -matrix that describes D can be extended to a complete weakly sign symmetric Po -matrix. This paper investigates the problem of completing weakly sign symmetric Po -matrices. It demonstrates that partial matrices representing all directed graphs of order 5 with edge strengths from 0 to 5 can indeed be completed to a weakly sign symmetric Po -matrix. Moreover, we established digraph characteristics that the partial Wss Po- matrices specifying digraphs of order 5 with up to 5 arcs which have a clique and are cyclic or acyclic have zero completion into a Wss Po- matrix. Insights gained from this class of matrix could be applied to fill gaps in data surveys, and business analytics by analysing complex relationships, allocating resources, network modelling, optimizing processes and managing risks.
Keywords: Cyclic digraphs, acyclic digraphs, clique digraphs, Matrix completion, Digraph, weakly sign symmetric Po-matrix