ALMOST-ALL TREES SHARE A COMPLETE SET OF IMMANANTAL POLYNOMIALS

Let chi be an irreducible character of the symmetric group S(n). For an n-by-n matrix A = (a(ij)), define [GRAPHICS] If G is a graph, let D(G) be the diagonal matrix of its vertex degrees and A(G) its adjacency matrix. Let y and z be independent indeterminates, and define L(G) = yD(G) + zA(G). Suppose t(n) is the number of trees on n vertices and s(n) is the number of such trees T for which there exists a nonisomorphic tree T such that d(chi)(xI - L(T)) = d(chi)(xI - L(T)) for every irreducible character chi of S(n). Then lim(n-->infinity)s(n)/t(n) = 1. (C) 1993 John Wiley & Sons, Inc.