 A forward, Graham scored on his full debut, a 6-1 victory over KR Reykjavik on 14 September 1964 in a European Cup preliminary round 2nd leg tie at Anfield.He scored a hat-trick in his first league game against Aston Villa 12 days later on 26 September, the game finished 5-1 to the Reds with Graham's goals coming in the 6th, 64th and 86th minutes.Graham was at Liverpool at an unfortunate time for himself as he found it difficult to dislodge the successful strike partnership of Ian St John and Roger Hunt and found first team appearances hard to come by.