Laplacian immanantal polynomials and the GTS poset on trees

Let T be a tree on n vertices with Laplacian LT and let GTSn be the generalised tree shift poset on the set of unlabelled trees on n vertices. Inequalities are known for coefficients of the characteristic polynomial of LT as we go up the poset GTSn. In this work, we generalise these inequalities to the q-Laplacian LTq of T and to the coefficients of all immanantal polynomials.