The left back joins Vale following his departure from League One champions Sheffield United at the end of the season.
Graham rose through the ranks at Sheff Utd and at the age of 18, he made his debut for the Blades against Coventry City at the end of the 2015/16 season.
Kelly added another first team appearance during the 2016/17 Checkatrade Trophy competition.
At the start of the 2016/17 campaign, Kelly went on loan to Stalybridge Celtic to get some more game time under his belt.
Elsewhere, the 19 year-old has represented his country at under-18 and under-19 level, earning four international appearances in each age category.
Kelly scored his first international goal at under-19 level against Latvia back in November 2015.
The defender becomes the sixth summer signing under Michael Brown and the fifth this week, following Rob Lainton, Cristian Montano, Joe Davis and Sam Hornby.
Graham will link up with his new team mates at the end of June when they return for pre-season ahead of the Sky Bet League Two campaign.
Stay tuned to the club website and official social media channels to hear more from Kelly over the coming days.