|Date of birth||29 July 1939|
|Place of birth||Glasgow, Scotland|
|1967||→ Toronto City (guest)||11||(1)|
|* Senior club appearances and goals counted for the domestic league only|
McGraw started his playing career with Greenock Morton. Playing in Division Two he set a Scottish record for most goals scored in a season. He finished as Morton's top goalscorer in five consecutive seasons, and appeared in their 1963 Scottish League Cup Final defeat. He helped the club to promotion to the top flight in 1964, their first time at that level since 1952. Two seasons later in 1966 the Ton were relegated.
McGraw left Morton for Hibernian, where he scored a goal which meant that the club reached a League Cup final. He took a number of pain killing injections in order to play while injured. This ruined his knees, causing great pain and necessitating the use of walking sticks for the rest of his life. He later played for Linfield for one season and spent a season back at Morton without appearing in a league match.
McGraw returned to Morton as manager in 1985. Players he managed included Derek McInnes, John Anderson, David Wylie and Alan Mahood. Towards the end of his time as manager, McGraw's team that missed promotion to the Scottish Premier Division by one goal in the 1995-96 season.
- Scottish First Division champions: 1986–87
- Scottish Second Division champions: 1994–95
- Scottish First Division Manager of the Year: 1986–87
- Scottish PFA Second Division Manager of the Year: 1994–95
- Scottish PFA Special Merit Award: 1998
- Graeme Ross (2004), Morton Greats, Breedon Books
- "A teacher who learned the hard way". HeraldScotland. Retrieved 25 January 2019.
- "Former Linfield striker is inducted into the Scottish Football Hall of Fame". Belfast Live. 19 October 2017. Retrieved 17 May 2019.
- "Allan McGraw". GMFC. Retrieved 17 May 2019.
- "Scottish Football Press release regarding Hall of Fame award for Allan McGraw". Linfield FC. 18 October 2017. Retrieved 17 May 2019.
- "Nicholl is voted top manager". The Glasgow Herald. 15 May 1995. Retrieved 23 July 2018.
- "Brown goes into World Cup extra time". Herald Scotland. 27 April 1998. Retrieved 15 May 2019.