  Let $K$ be a proper (i.e., closed, pointed, full convex) cone in ${\Bbb
R}^n$. An $n\times n$ matrix
  $A$ is said to be $K$-primitive if there exists a positive integer $k$ such
that $A^k(K \setminus \{0 \}) \subseteq$ int $K$; the least such $k$ is
referred to as the exponent of $A$ and is denoted by $\gamma(A)$. For a
polyhedral cone $K$, the maximum value of $\gamma(A)$, taken over all
$K$-primitive matrices $A$, is denoted by $\gamma(K)$. It is proved that for
any positive integers $m,n, 3 \le n \le m$, the maximum value of $\gamma(K)$,
as $K$ runs through all $n$-dimensional polyhedral cones with $m$ extreme rays,
equals $(n-1)(m-1)+1$ when $m$ is even or $m$ and $n$ are both odd, and is at
least $(n-1)(m-1)$ and at most $(n-1)(m-1)+1$ when $m$ is odd and $n$ is even.
For the cases when $m = n, m = n+1$ or $n = 3$, the cones $K$ and the
corresponding $K$-primitive matrices $A$ such that $\gamma(K)$ and $\gamma(A)$
attain the maximum value are identified up to respectively linear isomorphism
and cone-equivalence modulo positive scalar multiplication.
