Spectra with Positive Elementary Symmetric Functions

It is well known that if all the elementary symmetric functions of the eigenvalues of an n–by–n matrix are positive, then all its eigenvalues lie in the region of the complex plane {z : −π + πn < argz < π − π n}. Let A denote an n–by–n matrix with all diagonal entries nonzero and for which the length of the longest cycle in its directed graph is k, 2 ≤ k ≤ n. If, in addition, all the cycles in the directed graph of −A are signed negatively, then the elementary symmetric functions of the eigenvalues of A are positive and we ask whether its eigenvalues lie in the region {z : −π+ πk < argz < π− π k }. This is known to be true when k = 2 (sign–stability) and we prove it here for k = n − 1. We provide a counterexample for the case k = n − 3 and discuss related questions for more general classes of matrices with restricted length of the longest cycle.