Minimal exponent for transition matrix to be positive

Question

For an irreducible Markov Chain on a finite state space we have that it is aperiodic if and only if there is an integer $k$ such that the transition matrix satisfies $P^k>0$, i.e., all entries of $P^k$ are positiv. Assuming that a chain is aperiodic gives rise to the question which is the minimal $k$ such that the above mentioned property of $P$ is satisfied. Are there results on this simply from a matrix point of view? Preferably without using eigenvalues. I only know the minimal number of non-negative elements per line.