  Oscillatory matrices were introduced in the seminal work of Gantmacher and
Krein. An $n\times n$ 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.
