For graduation, the last thing I want to burden myself is to carry a handheld handbag/clutch. Instead, I would carry a SMALL shoulder bag or a cross body messenger bag to hold the "essentials" that help me stay comfortable during the long and uncomfortable process under a hot sun (usually). Examples of the essentials are: a paper fan (or the small plastic one that runs on battery), sunglasses, a water bottle, and a cell phone. A cell phone is a must, because you need to find your parents at the end of the ceremony when there is usually kaos from excitement and eagerness to join loved ones.
Make sure you wear very light clothing, because it can be extremely hot and stuffy under the long and breatheless gowns. I heard one year some girl wore a bikini underneath her gown to stay cool. Every graduate who found out about the incident praised this girl for being smart.