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

TL;DR

This paper introduces a generalized matrix function framework for tree Laplacians using symmetric functions, extending known inequalities within the GTS poset to broader classes of symmetric functions.

Contribution

It generalizes inequalities of matrix polynomial coefficients of tree Laplacians by employing arbitrary symmetric functions, expanding previous results involving Schur functions.

Findings

Established inequalities for generalized matrix polynomial coefficients using monomial and forgotten symmetric functions.

Extended the GTS poset inequalities to a broader class of symmetric functions.

Provided a unified framework linking symmetric functions and tree Laplacian polynomials.

Abstract

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.