  Let m and r be two integers. Let G be a connected r-regular graph of order n
and k an integer depending on m and r. For even kn, we find a best upper bound
(in terms of r and m) on the third largest eigenvalue that is sufficient to
guarantee that G has a k-factor. When nk is odd, we give a best upper bound (in
terms of r and m) on the second largest eigenvalue that is sufficient to
guarantee that G is k-critical.
