Graham attended Pulaski County High School in Dublin, Virginia, graduating in 1996. He holds many of the school's kicking records including most field goals in career (28, from 1992-1995), most field goals in a game (15, 1995), and longest field goal, a 54-yarder against Anacostia in 1995.
Graham played college football at Virginia Tech and was named to the first-team All-Big East Conference in all four seasons. In 1999, as a senior, Graham earned Big East Special Teams Player of the Year honors after leading the conference and breaking the school's single-season scoring record with 107 points on 56-of-57 extra points and 17-of-22 field goals. He was 68-of-93 (73.1%) in field goals for his Virginia Tech career and set a Virginia Tech and Big East record with 97 consecutive successful extra points. He left Virginia Tech as the all-time scoring leader in school history and Big East history with 371 points.