Dear member,
Did you issue the appointment letter to the employee? If yes, then tell the employee to adhere to the terms and conditions of the separation mentioned in the appointment letter.
If the appointment letter is not issued, then the terms and conditions of the separation are not binding on him/her.
In the meanwhile, do not register the employee's name on the PF portal. Show the employee as a casual employee and clear the payment for a week or so.
However, the larger issue is why the employee felt the need to resign in just a week. What were the compelling reasons? Does he/she consider a misfit in the company's culture? Did the HOD treat him/her harshly? Is it that the employee got a better opportunity? Rather than procedural norms for the separation, these are the issues your company needs to ponder.
Thanks,
Dinesh Divekar