On the exponent of several classes of oscillatory matrices

Oscillatory matrices were introduced in the seminal work of Gantmacher and Krein. An n×n matrix A is called oscillatory if all its minors are nonnegative and there exists a positive integer k such that all minors of A^k are positive. The smallest k for which this holds is called the exponent of the oscillatory matrix A. Gantmacher and Krein showed that the exponent is always smaller than or equal to n−1. An important and nontrivial problem is to determine the exact value of the exponent. Here we use the successive elementary bidiagonal factorization of oscillatory matrices, and its graph-theoretic representation, to derive an explicit expression for the exponent of several classes of oscillatory matrices, and a nontrivial upper-bound on the exponent for several other classes.