Alongside Mark Wilson, Graham took charge of Thistle for the second time since the Glasgow club parted ways with Kris Doolan.
The departed former forward received a minute's applause by the home support in the ninth minute - the number he wore for the club - in the first game at Firhill since his dismissal earlier this month.
Swiftly, though, it was the current number nine receiving all the adulation.
The superb Logan Chalmers clipped in a lovely ball for Graham to tap in his 17th goal of the season.
The veteran, though, squandered a few opportunities while the Jags were ahead, including one where he headed against a post.
And Muirhead's sixth goal in nine games meant Thistle had to settle for a draw that takes them five points clear of Raith Rovers, who have two games in hand.