Distance matrix polynomials of trees

Let G be a finite connected graph. If x and y are vertices of G, one may define a distance function d_G on G by letting d_G(x, y) be the minimal length of any path between x and y in G (with d_G(x, x) = 0). Thus, for example, d_G(x, y) = 1 if and only if {x, y} is an edge of G. Furthermore, we define the distance matrix D(G) for G to be the square matrix with rows and columns indexed by the vertex set of G which has d_G(x, y) as its (x, y) entry. In this paper we are concerned with properties of D(G) for the case in which G is a tree (i.e., G is acyclic). In particular, we precisely determine the coefficients of the characteristic polynomial of D(G). This determination is made by deriving surprisingly simple expressions for these coefficients as certain fixed linear combinations of the numbers of various subgraphs of G.