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, on G by letting d&x, y) be the minimal length of any path between x and y in G (with d&, x) = 0). Thus, for example, d&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&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 tied linear combinations of the numbers of various subgraphs of G.