As we know that the complexity of diagonalization of a matrix should be N^3, where N is the dimension of the matrix.
In MKL, there are a lot of functions that can calculate just some of the eigenvalues.
I want to know that when we just calculate one eigenvalue, if the time cost is divided by N?
You cannot expect that the computation time will be N times lower for only one eigenvalue. Of course, some performance benefit can be expected, but you need to know that there're parts common for any number of eigenvalues (like reducing the matrix to triangular form) and which we cannot make faster.