Professor Graham received his medical training at the University of New South Wales (UNSW) and subspecialty training (Cardiorenal diseases) at St. Vincent’s and Sydney Hospitals before moving to the US in 1977. There he worked at Southwestern Medical School, Dallas (postdoctoral fellow and then Assistant Professor), the Massachusetts General Hospital and Harvard Medical School (Associate Professor in Medicine and Head, Cardiac Biochemistry Laboratory), the Massachusetts Institute of Technology (Visiting Professor, Laboratory of HG Khorana) and the Cleveland Clinic Foundation (Robert C. Tarazi Professor and Chairman, Department of Molecular Cardiology) and Case Western Reserve University (Professor of Physiology and Biophysics).

He returned to Australia in 1994 as the inaugural director, Victor Chang Cardiac Research Institute, and Des Renford Professor of Medicine, UNSW. He was elected to Fellowship of the Australian Academy of Science in 2002, received a Centenary Medal (2000) and an Order of Australia (2009) for services to molecular cardiology, and is currently President, Association of Australian Medical Research Institutes; Member, Research Committee and Australia Fellowships Selection Committee, National Health and Medical Research Council, and founding Board Member and contributing scientist of EngeneIC Ltd. 

Professor Graham pioneered studies into the structure, function and signaling of adrenergic and other G-protein-coupled receptors. More recently, he has been involved in studies investigating the therapeutic potential of stem cells for cardiac diseases, and of regenerative mechanisms in the adult mammalian heart. His principal role in the Stem Cells Australia initiative will be to direct research efforts on the regenerative potential of cardiomyocyte and stem cell division and terminal differentiation regulated by c-kit.