Homogeneous linear recurrence relations of the determinants of distance matrices of trees
Zhiqi Liu, Hui Zhou
TL;DR
The paper develops a systematic subtree-decomposition framework for distance matrices of trees and derives a hierarchy of homogeneous linear recurrence relations for $\det(D_n)$, extending the classical Graham–Pollak result. By decomposing a tree into a small subtree $S_m$ and a remainder $R$ and considering all non-equivalent placements of the common vertex, the authors obtain explicit $m$-term recurrences for $4\le m\le 7$, including four-, five-, six-, and seven-term relations. These recurrences, together with suitable initial conditions, yield new proofs of Graham–Pollak’s formula and illuminate how different subtree configurations influence the determinant structure. The results provide a rich catalog of recurrence relations and raise questions about extending the method to larger subtrees ($m>7$) and the potential role of paths in maximizing the recurrence length.
Abstract
In 1971, by induction on $n$ and using a two-term linear recurrence relation, Graham and Pollak got a beautiful formula $$\det(D_n)=-(n-1)(-2)^{n-2}$$ on the determinant of distance matrix $D_n$ of a tree $T_n$ on $n$ vertices. The recurrence relations are very crucial when proving this formula by inductive method: in 2006, Yan and Yeh used two-term and three-term recurrence relations; in 2020, Du and Yeh used a homogeneous linear three-term recurrence relation. In this paper, we analyze the subtree structure of the tree and find four-term, five-term, six-term and seven-term homogeneous linear recurrence relations on $\det(D_n)$, as a corollary new proofs of Graham and Pollak's formula can be given.