Another simple proof of Graham and Pollak's theorem

Abstract Let T be a tree with n vertices, and D n be the distance matrix of T . Graham and Pollak (1971) discovered an elegant formula for the determinant of D n : det ( D n ) = − ( n − 1 ) ( − 2 ) n − 2 . It is surprising that it depends only on the order of T , not on the specific structure of T . By virtue of the classical Dodgson’s determinant-evaluation rule, Yan and Yeh (2006) presented a simple proof of the formula above. In this note, we give another simple proof, based on a homogeneous linear three-term recurrence relation: det ( D n ) + 4 det ( D n − 1 ) + 4 det ( D n − 2 ) = 0 .