Laplacian Immanantal polynomials and the GTS poset on Trees

TL;DR

This paper extends known inequalities for Laplacian characteristic polynomial coefficients of trees to the more general $q$-Laplacian and all immanantal polynomials, within the framework of the GTS poset.

Contribution

It generalizes inequalities from the Laplacian characteristic polynomial to the $q$-Laplacian and all immanantal polynomials for trees in the GTS poset.

Findings

Inequalities hold for coefficients of the $q$-Laplacian's immanantal polynomials.

Results extend the known inequalities from the Laplacian to a broader class of polynomials.

The work applies to the structure of the GTS poset on trees.

Abstract

Let $T$ be a tree on $n$ vertices with Laplacian $L_T$ and let $GTS_n$ 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 $GTS_n$. In this work, we generalize these inequalities to the $q$-Laplacian $L^q_T$ of $T$ and to the coefficients of all immanantal polynomials.