Generalized Matrix polynomials of Tree Laplacians indexed by Symmetric functions and the GTS poset

Let $T$ be a tree on $n$ vertices with $q$-Laplacian $L_T^q$ and Laplacian matrix $L_T$. Let $GTS_n$ be the generalized tree shift poset on the set of unlabelled trees on $n$ vertices. Inequalities are known between coefficients of the immanantal polynomial of $L_T$ (and $L_T^q$) as we go up the poset $GTS_n$. Using the Frobenius characteristic, this can be thought as a result involving the schur symmetric function $s_{\lambda}$. In this paper, we use an arbitrary symmetric function to define a {\it generalized matrix function} of an $n \times n$ matrix. When the symmetric function is the monomial and the forgotten symmetric function, we generalize such inequalities among coefficients of the generalized matrix polynomial of $L_T^q$ as we go up the $GTS_n$ poset.