Do you have to notify your past university of a name change (marriage)?

I graduated as Ms. A from the University. I then married after I received my degree, and became Mrs. B. Do I need to notify my University of the name change so that when prospective employers inquire of my information there is no mix up with the name? How does that work?

I just don't want an employer calling to verify my degree, asking about Mrs. B, to have the University say there was no Mrs. B, because they only have Ms. A on file. Do you have to notify your past university of a name change (marriage)?
Yes, you should notify your university for the exact reason you stated.