After making his Test debut against Tonga at the Sydney Cricket Ground in 1973, the tough prop developed a reputation as a solid technical scrummager and was a preferred starter in the early days of coach David Brockhoff's​ "step forward" era, which became a call to arms for the Australian forwards, in particular, to counter aggression with aggression in the 1970s.

Graham also captained NSW, Sydney Rugby Union and St George rugby clubs and was a stalwart of the St George rugby community, serving as a patron of the St George Sub District Rugby Club to this day.

Off the field, Graham made an enormous contribution to rugby over more than three decades, as president and chairman of the NSW Rugby Union and an executive council member of the International Rugby Board. Elected ARU chairman in 2005, he guided the code through an enlarged Tri Nations series and the expansion of the Super Rugby competition to 14 teams.

His many years of service - and his time as a player - are recognised in the Queen's Birthday honours list by becoming a member of the Order of Australia (AM) for "significant service to rugby union football as an administrator at the state, national and international level, as a player, and to the community".

In 2014 the NSWRU recognised Graham's contribution to rugby in the state with honorary life membership.

Graham currently serves on the board of the Sydney Cricket Ground Trust and is a director of the Weary Dunlop Foundation.