Graham Bowkett was born in Pontypridd and signed from Pontypridd at the start of the 1955/56 season, he played as a forward in the 2nd pre-season practice game on the 15/8/55.

In in the Summer of 1955 he was playing in the GB Youth team on a tour of Germany, and had been wanted by Cardiff City. However he choose City. 

He made his Southern League debut against Yeovil on the 27th of August 1955 at Yeovil. He played for throughout the season for the reserves scoring once. 

Graham played for the 1st team, as a forward in the delayed 1954/55 Devon Bowl Final on the 8/11/1955 against Torquay at home, in a 2-2 draw.

Because of National Service and being called up, Exeter City thought they had lost his services and despite a quick promotion to Lance corporal, he was given permission to play and often joined the reserves from his camp in Devizes. Making a total of 18 apperance in 1955/56, scoring once.

He therefore played for the reserves in 1956/57 and managed to make 10 apperances and played again as a full-back, in the 1957/58 season in the Southern League.