The ancestral matrix of a rooted tree

Given a rooted tree T with leaves v ₁ ,v ₂ ,…,v _n , we define the ancestral matrix C(T) of T to be the n×n matrix for which the entry in the i-th row, j-th column is the level (distance from the root) of the first common ancestor of v _i and v _j . We study properties of this matrix, in particular regarding its spectrum: we obtain several upper and lower bounds for the eigenvalues in terms of other tree parameters. We also find a combinatorial interpretation for the coefficients of the characteristic polynomial of C(T), and show that for d-ary trees, a specific value of the characteristic polynomial is independent of the precise shape of the tree.