The Inverse Characteristic Polynomial Problem for Trees

Abstract

It is known that any real polynomial is attained as the characteristic polynomial of a real combinatorially symmetric matrix, whose graph is either a path or a star. We conjecture that the same is true for any tree (this is so for complex characteristic polynomials and complex matrices). Here, we constructively prove a very large portion of this conjecture by a method that mates the graph of the polynomial with a notion of balance of the tree relative to a choice of root for the tree. Included is the first constructive proof for the case of the path, as well as the case of any tree on fewer than ten vertices. It also includes the known case of polynomials with distinct real roots for any tree (in a new way). This work is motivated by, and lies in contrast to, the considerable study of possible multiplicity lists for the eigenvalues of real symmetric matrices, whose graph is a tree.