Laplacian Immanantal polynomials and the GTS poset on Trees

Let $T$ be a tree on $n$ vertices with Laplacian $L_T$ and let $GT{S}_n$ be the generalized tree shift poset on the set of unlabelled trees on $n$ vertices. Inequalities are known for coefficients of the characteristic polynomial of $L_T$ as we go up the poset $GT{S}_n$. In this work, we generalize these inequalities to the $q$-Laplacian $L_T^q$ of $T$ and to the coefficients of all immanantal polynomials.