Almost principal minors of inverse M-matrices

It is well known that if an inverse M-matrix has a 0 entry, then it must be reducible and thus have many more 0 entries. This property is actually a special case of a deeper phenomenon that might be loosely described as relations among vanishing almost principal minors in an inverse M-matrix. This phenomenon encompasses both minors of nested dimension (a certain loose monotonicity) and minors of the same size in loosely related positions. This phenomenon is limited to almost principal minors and, where possible, converses and examples are given to show the limit of the extent of this phenomenon. It is also shown that if one almost principle minor is contained in another, then the magnitude of the former is larger than that of the latter.