On a conjecture of Graham and Lovász about distance matrices

In their 1978 paper \Distance Matrix Polynomials of Trees", [4], Graham and Lov asz proved that the coeÆcients of the characteristic polynomial of the distance matrix of a tree (CPD(T )) can be expressed in terms of the numbers of certain subforests of the tree. This result was generalized to trees with weighted edges by Collins, [1], in 1986. Graham and Lov asz computed these coeÆcients for all trees on less than 8 vertices, noticed that the sequence of coeÆcients was unimodal with peak at the center, and conjectured that this was always true. In this paper, we disprove the conjecture. The coeÆcients for a star on n vertices are indeed unimodal with peak at h n 2 i , but the coeÆcients for a path on n vertices are unimodal with peak at n(1 1= p 5). Adjacency matrices have received a lot of attention, particularly with respect to their characteristic polynomials (CP 's). It has only been in the last few years that very much work has been done with distance matrices. This is partly because adjacency matrices have many more zeroes than distance matrices, and therefore it is easier to compute their CP 's, and partly because even adjacency matrices themselves are not well understood. When the rst pair of trees (surely the simplest kind of graphs) were found that had the same CP of adjacency matrices, it was shown that we could not so easily nd a mapping from the set of all trees with n vertices to the set of polynomials of degree n. Later this desire lead to the discovery of two trees with the same CP of distance matrices [6]. If only each tree could have associated with it some unique polynomial; for then the set of trees would somehow be like an algebra, and might yield information from algebraic techniques. Unfortunately, at the moment there are no good candidates for such an