Asian Journal of Pure and Applied Mathematics

A square matrix is a Wss P_o-matrix if the off-diagonal elements have the property that if the element at position (i, j) is not zero (i ≠ j), then the element at position (j, i) must either have the same sign or be zero. A Wss P_o -matrix is considered to have positionally symmetric cycle if its entries are symmetric with respect to their positions in the matrix. A digraph D has a Wss P_o-matrix completion if every partial weakly sign symmetric P_o -matrix that describes D can be extended to a complete weakly sign symmetric P_o -matrix. This work extends our earlier investigation into the completion of weakly sign-symmetric Po-matrices, focusing on digraphs of order 5 with up to 5 arcs. In that study, we proved that every acyclic or cyclic digraph of order 5 with up to 5 directed edges can be completed into Wss P_o-matrix. Our research therefore, advances our previous findings by examining digraphs that contain positionally symmetric cycles. Our goal is to further identify and characterize the structural properties of such digraphs that leads to completion or non-completion. Our study established that directed graphs with 5 vertices and either 4 or 5 directed edge which contain a positionally symmetric cycle do not have completion into a Wss P_o-matrix. Moreover, we observed that digraphs with 5 arcs and positionally symmetric cycles inherit the non-completion property from the corresponding 4-arc digraphs with the same cycle structure. These findings could be applied in practical problems such as studying relationships in networks, filling missing data, and solving optimization tasks.