The 22-year-old has signed a one-year deal, while goalkeeper Colin Stewart is currently weighing up an offer.
Dominic Shimmin, Donovan Simmonds, Alex Walker and Alan Reid, meanwhile, have been informed their contracts will not be renewed.
Those four join Stevie Masterton and Kevin Finlayson in the roster of players whose futures look to lie away from Cappielow.
However, the club will submit the released players' names to the new manager in case he is interested in re-signing them.
Ryan McGuffie is on his way to Queen of the South, Jim McAlister is likely to sign for a Premier League club and Erik Paartalu has switched to Brisbane Roar in Australia.
Graham, right, made 26 appearances last season but only managed two goals in a campaign undermined by injury.
The striker is looking forward to making a major impact next season. He said: "I'll be looking to try my hardest in pre-season and hit the ground running. I am only 22 and am still learning every day. When I was on loan at East Stirling in the 2008-09 season I felt I really improved as a player.
"When I came back I was flying, but unfortunately injured my ankle ligaments. Hopefully, I can steer clear of injuries and really show what I am capable of. I scored 19 when I was at East Stirling for a season and I believe I have the ability to get into double figures at Morton. I had a few options, but I am glad to be staying here."
Iain Russell is confident of remaining at Ton after positive talks with the club. The striker had a loan spell at Stirling Albion last year under the stewardship of Allan Moore, who has been linked with the Ton job.
Ton chairman Douglas Rae has already stated the club will take their time over finding a new manager, with the end of the month a likely timescale for any appointment.