Brought up in the Castlemilk area of Glasgow, Arthur Graham began his career with Cambuslang Rangers. At that time Bobby Calder was the Aberdeen scout that had the knack of taking some of the most promising talent in the country to Pittodrie right from under the noses of the Old Firm. Arthur Graham was no different as Bobby used all of his persuasive powers to take Graham to Aberdeen. After signing in February 1970 Graham made his first team debut against Dunfermline on 21st March that year, a week after Aberdeen had defeated Kilmarnock to reach the Scottish Cup Final. Four days later and Graham was in the Aberdeen side that shocked Celtic at Parkhead in what was a cup final rehearsal. Aberdeen won 2-1 and Arthur set up the Dons first goal before scoring the winner himself. 21 days after making his debut. 'Bumper' as he became known was thrust into the Aberdeen team for the cup final against Celtic at Hampden before a 108,000 crowd.
Graham was never overawed by the occasion and played his part in the Dons memorable 3-1 win. After the game Arthur made a lovely gesture by offering scout Bobby Calder his winners medal. Bobby turned down the offer but was clearly touched by the young wingers gesture. Arthur Graham went on to become a feature in the Aberdeen side and he was in the Aberdeen team that won the League Cup in 1976. He was capped at U-23 level before his international career was halted after an incident in Copenhagen in 1975. A subsequent ban was later lifted and Arthur Graham made his full international debut against East Germany in 1977. By that time Arthur had moved to Leeds United in a £125,000 transfer despite the best efforts of new Dons boss Billy McNeill to persuade him to stay.
Arthur Graham had played more than 300 games for the Dons scoring 34 goals. He was certainly the finished article by the time he played in England and he was a huge favourite at Elland Road where he once scored a hat trick in six minutes against Birmingham. After leaving Leeds he was a surprise signing for Manchester United where he enjoyed one full season at Old Trafford in the first team. Arthur still remains in the game coaching with his old friend Jimmy Lumsden in the Leeds area.