After World War II, Graham studied at the University of Edinburgh and Moray House College, Edinburgh, during which time he was active in literary circles and co-edited a volume of Scottish student verse.
He returned to Shetland to take up a teaching post and taught at the Anderson Educational Institute and the Scalloway Junior High School.
He contributed biographical sketches to the important textbook The Shetland Book (1967), edited by Andrew T. Cluness.
The text included a translation into Danish by Martin Melsted of an article by William J. Tait on Shetland language and literature.
In 2000, the Shetland Library published his selected poems in Love's Laebrack Sang, a volume which demonstrates Graham as both a committed political poet, engaged in making response to the public events of the day via satire, and a Shetland poet with a deep love of his community and its history.