After leaving college with a degree in Applied Biology, Graham joined the Department of Clinical Endocrinology at the Imperial Cancer Research Fund (ICRF) in 1976. He remained in the department until 1991 when he re-located to an ICRF sponsored laboratory at the Radcliffe Infirmary in Oxford.
In 1995 Graham returned to the ICRF to establish a core facility known as the Equipment Park. In 2002 the ICRF merged with Cancer Research Campaign to form Cancer Research UK which in turn became part of the Francis Crick Institute at Kings Cross. Graham moved the facility into the Crick in 2016, now renamed the Genomics Equipment Park.