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Let $T$ be a tree on $n$ vertices with $q$-Laplacian $L_T^q$ and Laplacianmatrix $L_T$. Let $GTS_n$ be the generalized tree shift poset on the set ofunlabelled trees on $n$ vertices. Inequalities are known between coefficientsof 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 resultinvolving the schur symmetric function $s_{\lambda}$. In this paper, we use anarbitrary symmetric function to define a {\it generalized matrix function} ofan $n \times n$ matrix. When the symmetric function is the monomial and theforgotten symmetric function, we generalize such inequalities amongcoefficients of the generalized matrix polynomial of $L_T^q$ as we go up the$GTS_n$ poset.

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