After graduating in Chemistry in 1980 from the University of Oxford, Graham worked in the Science Faculty of the Open University for three years. He joined the central administration at Cambridge in 1983 and held a number of posts, including Secretary of the School of the Humanities and Social Sciences. In 1996 he became Deputy Secretary General of the Faculties and subsequently Academic Secretary of the University which he held until 30 September 2015. In this role he was Secretary of the General Board and Head of the Academic Division with wide ranging responsibilities for the University's teaching and research and for academic strategy.
He has served as a Governor of Bromsgrove School, and of the Papworth NHS Foundation Trust. He is currently a Trustee/Director of the Babraham Institute and a member of the University's Board of Scrutiny.