Factoring distance matrix polynomials

In this paper we prove that a vertex-centered automorphism of a tree gives a proper factor of the characteristic polynomial of its distance or adjacency matrix. We also show that the characteristic polynomial of the distance matrix of any graph always has a factor of degree equal to the number of vertex orbits of the graph. These results are applied to full k-ary trees and some other problems. Adjacency matrices and their spectra have arisen naturally as a tool with which to study graphs. The idea of a distance matrix seems a natural generalization, with perhaps more speci city than that of an adjacency matrix. However, the spectra of neither adjacency matrices nor distance matrices characterizes even trees; see [12]. Distance matrices and their spectra have also arisen independently from a data communication problem studied by Graham and Pollack, [9], in 1971, in which their most important feature is their number of negative eigenvalues. See Graham and Lov asz, [8], or [4] for a description of the problem. Let G be a graph. Typically, the computation of the characteristic polynomial of the adjacency matrix of a graph G, CPA(G), has been related to nding certain kinds of subgraphs of the original graph. For instance, let T be a tree on n vertices. Then the each coeÆcient of CPA(T ) can be computed knowing only the number of matchings in the tree with a xed number of edges. Graham and Lov asz have proven a similar theorem for distance matrices of trees, in which each coeÆcient of CPD(T ) can be computed using linear combinations of the numbers of certain subforests with three di erent xed numbers of edges, see [8]. The coeÆcients of the linear combinations are related to the degrees of vertices in the subforests. Thus it is much more diÆcult to compute these coeÆcients for distance matrices than for adjacency matrices. In this paper we present a technique for nding proper factors of the characteristic polynomial of the distance (or adjacency) matrix of a tree. This can greatly simplify the process of nding the eigenvalues of the distance matrix. The method is to look for linearly independent vertex-centered automorphisms of the tree, and to include the information from the vertex orbits of the tree. Usually the obvious automorphisms and the vertex orbit component together are enough to multiply to the correct degree and we have the whole characteristic polynomial. We use this method to compute the characteristic polynomials of the distance matrices of some examples. We also provide a lower bound on the number of times that 2 is an eigenvalue of the characteristic polynomial of a tree, which is an improvement of [13]. Finally, we compute the characteristic