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.
Such was the dominance of both St John and Hunt he missed out on the 1965 FA Cup Final win and only made 1 appearance during the Reds run to the league title the following season, obviously, not enough to gain him a medal.
The 1969–70 season was the most successful for him in a red shirt, with Hunt and St John both pensioned off he made 54 (ever present) first-team appearances and scored 21 goals which included a stunning strike in the Merseyside derby at Goodison Park, the same game that saw Everton's Sandy Brown score 'that' infamous own goal.
Shortly afterwards, though, two new strikers arrived in the form of Kevin Keegan from Scunthorpe United and John Toshack from Cardiff City, Graham lost his place once again.
[2] After 132 games and 37 goals Graham joined Hamilton Academical in 1977 for a, then, record transfer fee of £15,000, he, again, made over 100 appearances scoring 42 times.