Distance matrix polynomials of trees

Abstract

For a finite undirected tree T with n vertices, the distance matrix D(T) of T is defined to be the n by n symmetric matrix whose (i, j) entry is d_ij, the number of edges in the unique path from i to j. Denote the characteristic polynomial of D(T) by

$$\Delta _T (x) = det (D (T) - xI) = \sum\limits_{i = 0}^n {\delta _k } (T)x^k .$$

In this talk we describe exactly how the coefficients δ_K(T) depend on the structure of T. In contrast to the corresponding problem for the adjacency matrix of T, the results here are surprisingly difficult, requiring the use of a number of interesting auxiliary results.