Here is the list of some samples of cover letter for Graduate Position Applications from various universities. It would be helpful to have a look at these and make any adjustment to your own application. Good luck !!!

University of Louisville: post-graduate cover letter sample

University of New South Wales (Australia): Engineering graduate position cover letter sample

University of Virginia: CV and Cover Letter Writing for Academic & Research Audiences instruction with samples

Claremont Graduate University: sample academic cover letter

Purdue University: academic cover letters

General resources:

MIT: Instructions on how to write cover letters with sample

The Pennsylvania State University: resume and cover letter tips

Arizona State University: cover letter tips and samples