Undrafted, Mink originally played
high school hockey with Stowe High School, winning the Division III state championship in his sophomore year and Division II state championship in his senior year. After high school, he attended Northfield Mount Hermon school for a post graduate year. He then was a recruited walk on to play collegiate hockey with the
University of Vermont in the ECAC. After an unimpressive freshman year, Mink gradually improved within the Catamounts and in his junior year placed second on the team in scoring with 17 goals and 29 points in 32 games. Just prior to his senior year in 2001–02, Mink left Vermont and turned pro, signing with the Portland Pirates on September 25, 2001. In his first professional season, Mink made his debut in the
ECHL, appearing in 29 games with the
Richmond Renegades before moving up to the Pirates. Following an impressive 34 points in 56 games, Mink was signed by their NHL affiliate, the
Washington Capitals for the
2002–03 season and continued to play for the Pirates and later on the
Hershey Bears, helping them as an assistant captain to the 2006 and 2009
Calder Cups. On July 8, 2009, Graham signed a two-year free agent deal with the
Florida Panthers. He was then assigned to AHL affiliate, the
Rochester Americans where he served as an alternate captain for the
2009–10 season. On August 3, 2010, Mink was traded by the Panthers to the
St. Louis Blues in exchange for
T.J. Fast. On July 13, 2011, Mink left the
Peoria Rivermen and signed a one-year
American Hockey League contract to return to the
Hershey Bears for the
2011–12 AHL season. After 12 seasons of professional hockey in predominantly the AHL, Mink was signed to his first European contract, agreeing to a one-year deal with Dornbirner EC of the Austrian Hockey League on August 10, 2013. Mink played one season with Dornbirner, totaling 23 goals and 23 assists in 51 games before retiring from hockey. ==Personal==