In a primitive, stochastic matrix, the 2nd largest eigenvalue depends on which property of the elements of matrix?

Question

We know the largest eigenvalue of a primitive, stochastic matrix is 1. The 2nd largest eigenvalue is strictly smaller than 1. But can we get more information of 2nd largest eigenvalue from the elements of matrix? Something like this, "2nd largest eigenvalue depends on maximum/minimum elements of the matrix" or "The smaller the minimum element of the matrix is, the higher is the 2nd largest eigenvalue" or something like this. My examples may not correct, please provide correct information.