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
The Pennsylvania State University: resume and cover letter tips
Arizona State University: cover letter tips and samples