There is no word that comes to mind which exclusively applies to graduates and I do not think such a word would make sense. It would imply that accomplishing studies is inherently bad.
However there are words which would apply especially to graduates in particular contexts, since they are educated. These usually relate to the negative connotations associated with and symptoms exhibited by knowledge and learning zealots. Here is a somewhat prioritized list of suggestions:
Egghead seems the most applicable. It applies to a person who is only academically smart. It is especially applies to people whose head looks like an egg because they are old and balding after experiencing a life devoted to study.
"For a man who served in the military, Superintendent Chalmers can be such an egghead sometimes."
In reference to a schoolmaster or teacher, Pedant can refer to a person who makes a vain display of their learning, which is especially applicable when they exhibit trivial knowledge. This does necessitate that a person be considered somehow learned, as a graduate would be. The adjective form, pedantic, usually refers to the pejorative sense. More modern usage mistakes this as a word for people who are too picky with language specifically.
"Principal Skinner is soooo pedantic. He scolded me on the grammatical difference between the words 'can' and 'may' for hours."
Smartypants or Mr. Smartypants is used similarly to pedant, except without the implicit comparison to a teacher. I do not pretend to know why pants are involved and am not sure if I even want to learn.
"Principal Skinner even has a monthly subscription to Mr. Smarty Pants. Maybe that's why he acts like such a Smartypants."
Geek refers to somebody who is obsessive and knowledgeable about a particular subject, as somebody who holds a specialized college degree might be. However it can pertain to any subject, even if they are not considered academic.
"I would never dream of dating a geeky slob like Comic Book Guy!"
Nerd can be used to describe somebody who is excessively studious, which may be possible if the effort of study becomes an interference to other pursuits. This word is more likely to be ascribed to an undergraduate though, since graduation implies a cessation of studies.
"Stop being such a nerd, Milton!"
Bookworm may be used similarly to nerd with special emphasis on a reading addiction, although it is not always considered pejorative.
"Lisa is a bookworm."
Pedagogue may not be inherently pejorative but it can be a synonym for pedant. It originally referred to greek slaves in charge of caring for a master's children and taking them to school.
"Pedagogue Otto, at your service."
Poindexter is a name which is occasionally associated with nerdiness in reference to a Felix the Cat character utilizing it.
Hopefully despite being an imperfect match, you can find an (in)appropriate enough insult amongst this list of suggestions. Do have fun with your ad-hominem attacks!