I investigated how the Grahams got to Ireland, and put this together from several sources. I haven't confirmed all the "facts" below, but I thought it was worth posting as a signpost to some interesting research.
The origins of the Grahams seem to be largely Anglo-Saxon by origin. They lived on the border of England and Scotland, and were no angels. They had became "border reivers" (outlaws), raiding cattle and burning castles belonging to their sworn enemies. But became worn down by the ever increasing success of the goverment forces and their local allies, they accepted they would have to move if they ever were to have a settled life again.
Most present day Graham's in Ireland stem from an extended family of that name who settled in White Park County Antrim in the early 1600's, having been banished as reivers from Scotland where they suffered from persecution and certain death, as a law had been passed that anyone by the name of Graham were to be hanged summarily.
It is no coincidence that the most prevalent christian name of this Irish Graham Clan became James, after one of six brothers, James, the others being, Josias, Alexander, Gawn, Samuel, David who set about creating a Grahamstown at beautiful Glenwherry, County Antrim, some half way between Ballymena and Larne. The significance of the name and the perpetuating of it in every generation of the family no doubt hinted that this was a branch of the famous Graham Clan of Sir James Graham,(1612-1650).