Almost all trees are co-immanantal

Abstract

If T is a tree, denote by A( T) its adjacency matrix and by L( T) its Laplacian matrix. Write d 2 for the “second immanant” matrix function. The main result is this: Let t n be the number of trees on n vertices. Let s n be the number of such trees T for which there is a nonisomorphic tree T′ such that d 2( xI − A( T)) = d 2( xI − A( T′)) and d 2( xI − L( T)) = d 2( xI − L( T′)). Then s n / t n → 1 as n → ∞.