Another simple proof of Graham and Pollak’s theorem

Abstract

Let T be a tree with n vertices, and Dn be the distance matrix of T. Graham and Pollak (1971) discovered an elegant formula for the determinant of Dn: det(Dn)=−(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(Dn)+4det(Dn−1)+4det(Dn−2)=0.