when it comes Graphic Design field... where you graduated from is hardly an issue. It doesn't really matter if you're from community college or top 10 college. Graphic Design is all about how talented you are, not where you graduated from. As Reba said... your portfolio will be your most important asset. Employers do not really care what company you worked for, where you graduated from or what your GPA is or anything relating to academic because graphic design has nothing to do with it. It's a creativity field. They need to see your works.
one thing you have to expect.... Graphic Design field is very oversaturated and many people do not have full-time jobs... it's mostly contractual type. Creativity field is notoriously hard to make a decent living and that has always been like that for many years. that's why you always hear many stories about struggling writers/artists/musicians.
so I highly recommend you to expand your skills beside graphic design in order to stay afloat so that you can do other jobs while searching for graphic-related job. I have many many many friends who are graphic design major. they all do photography, logo design, web design, magazine design, etc.... and also odd jobs like waitressing, mr. fix-it, etc. very few of them are doing very well and they have their own businesses. so think carefully and plan well.