The distribution of eigenvalues of doubly cyclic Z+-matrices

It is shown that every matrix in a large class of n-by-n doubly cyclic Z+ matrices with negative determinant has exactly one eigenvalue in the closed left half-plane. This generalizes a result for n=4 used in a recent analysis of cancer cell dynamics. A further conjecture is made based on computational evidence. All work relates to the inertia of certain doubly cyclic circulants.