Malcolm Graham taught mathematics and engineering courses at Marion Military Institute in Marion, Alabama, during the 1948-49 academic year. From 1951-55, he was an assistant professor and later promoted to associate professor of mathematics at Longwood College in Farmville, Virginia. Longwood is one of the state colleges of Virginia with a strong emphasis on teacher training programs. In 1955-56, he was associate professor of mathematics at East Carolina University in Greenville, North Carolina.

From 1956 to 1985, Graham taught at UNLV, holding the rank of professor of mathematics since 1964. As of 1985, he held the longest tenure of any professor at UNLV (29 years).

In 1956, UNLV was known as the Southern Regional Division of the University of Nevada, Reno. Since there were no buildings, the faculty taught wherever there was space. Initially, Graham was the only full time member of the mathematics staff and taught mathematics classes in a Sunday School room of the Baptist Church in the Las Vegas High School and at Nellis Air Force Base. The school occupied the first campus building in 1957 and held its first commencement in 1964 with 29 graduates. In 1965, it became Nevada Southern University, and in 1969 the name was changed to the University of Nevada, Las Vegas.