Integrally normalizable matrices and zero-nonzero patterns

The problem of determining whether or not an n×n integer matrix is diagonally similar to an integer matrix with n−1 off-diagonal entries equal to 1 is studied. Such a matrix is called integrally normalizable, and a zero-nonzero pattern is integrally normalizable if each matrix with this zero-nonzero pattern is integrally normalizable with respect to the same set of n−1 off-diagonal entries. Matrices that are integrally normalizable with respect to a fixed spanning tree, and integrally normalizable zero-nonzero patterns are characterized. The maximum number of nonzero entries in an n×n integrally normalizable zero-nonzero pattern is shown to be n2+3n−22. Extensions to matrices over other integral domains are also presented.