The symmetric inverse M-matrix completion problem

Necessary and sufficient conditions are given on the data for completability of a partial symmetric inverse M-matrix, the graph of whose specified entries is a cycle, and these conditions coincide with those we identify to be necessary in the general (nonsymmetric) case. Graphs for which all partial symmetric inverse M-matrices have symmetric inverse M-matrix completions are identified and these include those that arise in the general (positionally symmetric) case. However, the identification of all such graphs is more subtle than the general case. Finally, we show that our new cycle conditions are sufficient for completability of all partial symmetric inverse M-matrices, the graph of whose specified entries is a block graph.