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\begin{document}

\title[Generalized Matrix polynomials of Tree Laplacians]{Generalized Matrix polynomials of Tree Laplacians indexed 
	by Symmetric functions and the $\GTS$ poset} 

\author{Mukesh Kumar Nagar}
\address
{Department of Mathematics and Statistics\newline \indent
Indian Institute of Technology Kanpur  \newline \indent
Kanpur 208 016, India}
\email{mukesh.kr.nagar@gmail.com}

\author{Sivaramakrishnan Sivasubramanian}
\address
{Department of Mathematics \newline \indent
	Indian Institute of Technology Bombay \newline \indent
	Mumbai 400 076, India}
\email{krishnan@math.iitb.ac.in}



\begin{abstract}
Let $T$ be a tree on $n$ vertices with Laplacian matrix $L_T$
and $q$-Laplacian $\sL_T^q$. 
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 $\sL_T^q$ as one moves 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 
$\sL_T^q$ as one moves up  the $\GTS_n$ poset.
\end{abstract}


\keywords {Tree, $\GTS_n$ poset, Laplacian, 
monomial symmetric function, generalized matrix polynomial}

\subjclass[2010]{05C05, 06A06, 15A15}

\maketitle

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\font\rms=cmr8 
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\markright{\its S\'eminaire Lotharingien de
Combinatoire \bfs 83 \rms (2021), Article~B83a\hfill}
%\def\thepage{}


\section{Introduction}
\label{sec:intro}


For a positive integer $n$, let $[n]=\{1,2,\ldots,n\}$ and  let
$\SSS_n$ denote the symmetric group on the set $[n]$.
We denote partitions $\lambda$ of the number $n$ as $\lambda \vdash n$. 
We write partitions using the exponential notation, with multiplicities of parts written as exponents.
For $\lambda \vdash n$, let $\chi_{\lambda}^{}$ be the irreducible 
character of $\SSS_n$ 
over $\CC$ indexed by $\lambda$. We think of $\chi_{\lambda}^{}$ 
as a function $\chi_{\lambda}^{} : \SSS_n \mapsto \ZZ$.
With respect to an irreducible character 
$\chi_{\lambda}^{}$, 
define the immanant of the $n \times n$
matrix $A = (a_{i,j})_{1 \leq i,j \leq n}$ as 
\begin{equation}
  \label{eqn:immanant}
  d_{\lambda}(A)  =  \sum_{\psi \in \SSS_n} \chi_{\lambda}^{}(\psi) 
  	\prod_{i=1}^n a_{i,\psi(i)}. 
\end{equation}

Let 
$\Lambda_{\QQ}^n$ denote the vector space of degree $n$ symmetric functions with 
coefficients from $\QQ$.  
The set of monomial symmetric functions $\{m_{\lambda} \}_{\lambda\vdash n}$ is one of the well known bases of $\Lambda_{\QQ}^n$. 
We refer the reader to the books by
Stanley \cite{EC2} and  by Mendes and Remmel \cite{mendes-remmel-book} for background on
symmetric functions.  $\Lambda_{\QQ}^n$ actually has an inner product structure as well. Another inner product space often studied is $\CF_n$,
the space of class functions from $\SSS_n \mapsto \QQ$.  Further, there is a well
known isometry between these two spaces called the Frobenius characteristic, denoted  
$\ch: \CF_n \rightarrow \Lambda_{\QQ}^n$
(see Stanley \cite{EC2} for more details).

Let $\gamma \in \Lambda_{\QQ}^n$ and consider $\Gamma_{\gamma} = \ch^{-1}(\gamma)$
its inverse Frobenius image.  Clearly, $\Gamma_{\gamma} \in \CF_n$ is a class function 
indexed by $\gamma$.
Define the generalized matrix function (GMF henceforth) of an $n \times n$ matrix $A$ with respect
to $\gamma$ as 
\begin{equation}
  \label{eqn:gen_matrix_fn}
  d_{\gamma}(A) = \sum_{\psi \in \SSS_n} \Gamma_{\gamma}(\psi) \prod_{i=1}^n a_{i,\psi(i)}.
\end{equation} 

Define the generalized matrix polynomial $\zeta_{\gamma}^A(x)$ of 
$A$ in a new variable $x$ with respect to a symmetric function $\gamma$ as 
follows:  $\zeta_{\gamma}^A(x)=d_{\gamma}(xI-A)$.  
It is well known  that the inverse Frobenius image $\ch^{-1}(s_{\lambda})$ 
of the Schur symmetric function $s_{\lambda}$ is $\chi_{\lambda}^{}$,
the irreducible character of $\SSS_n$ over $\CC$ indexed by $\lambda$  (see \cite{EC2}). 
Thus, from \eqref{eqn:immanant} and \eqref{eqn:gen_matrix_fn}, we can see that  $d_{s_{\lambda}}(A)=d_{\lambda}(A)$ and $\zeta_{s_{\lambda}}^A(x)=d_{\lambda}(xI-A)$.

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\markboth{\SMALL MUKESH KUMAR NAGAR AND SIVARAMAKRISHNAN SIVASUBRAMANIAN}{\SMALL GENERALIZED MATRIX POLYNOMIALS OF TREE LAPLACIANS}


 
Csikv{\'a}ri in \cite{csikvari-poset1} defined a poset 
on the set of unlabelled trees with $n$ vertices that we
denote in this paper as $\GTS_n$.  Among other results, he showed that 
if one moves up along the $\GTS_n$ poset, all coefficients of the 
characteristic polynomial of the Laplacian matrix $L_T$ of a tree 
$T$ decrease in absolute value.
This result was generalized by Nagar and Sivasubramanian in \cite{mukesh-siva-gts} to 
immanantal polynomials of $\sL_T^q$ indexed by any $\lambda \vdash n$, where $\sL_T^q$ is
the $q$-Laplacian matrix of $T$ (see Theorem~\ref{thm:main_earlier}).

Let $\RR^+$ denote the set of non-negative real numbers and $\RR^+[q^2]$ denote the 
set of polynomials in $q^2$ with coefficients in $\RR^+.$   Let $m_{\lambda}\in \Lambda_{\QQ}^n$ 
be the monomial symmetric function indexed by $\lambda \vdash n$. 
In Section~\ref{sec:gmfs_monomial}
of this paper, we prove the following result which shows monotonicity 
of the coefficient 
of $(-1)^rx^{n-r}$ for $0 \leq r \leq n$ in 
$\zeta_{m_{\lambda}}^{\sL_{T}^q}(x)$ when we move up along $\GTS_n$. 

\begin{theorem}
\label{thm:main}
Let $T_1$ and $T_2$ be two trees with $n$ vertices such that $T_2$ covers $T_1$ 
in $\GTS_n$.  Let $\sL_{T_1}^q$ and $\sL_{T_2}^q$ be the $q$-Laplacians 
of $T_1$ and $T_2$ respectively.  For $\lambda \vdash n$, let 
\begin{align*}
  \zeta^{\sL_{T_1}^q}_{m_{\lambda}}(x) &=  d_{m_{\lambda}}(xI -\sL_{T_1}^q) = \sum_{r=0}^n (-1)^r 
c_{m_{\lambda},r}^{\sL_{T_1}^q}(q) x^{n-r} \mbox{ and} \\
\zeta^{\sL_{T_2}^q}_{m_{\lambda}}(x) & =  d_{m_{\lambda}}(xI -\sL_{T_2}^q) = \sum_{r=0}^n (-1)^r 
c_{m_{\lambda},r}^{\sL_{T_2}^q}(q) x^{n-r}.
\end{align*}
Then for all $\lambda \vdash n$, we have  
$c_{m_{\lambda},r}^{\sL_{T_1}^q}(q) - c_{m_{\lambda},r}^{\sL_{T_2}^q}(q) 
\in \RR^+[q^2]$, where $0 \leq r \leq n$.   Further if $\lambda \neq 2^k,1^{n-2k}\vdash n$, 
then, $d_{m_{\lambda}}(xI -\sL_{T_1}^q)=d_{m_{\lambda}}(xI -\sL_{T_2}^q)=0.$
\end{theorem}


Recall that for $\gamma \in \Lambda_{\QQ}^n$, $\Gamma_{\gamma} = \ch^{-1}(\gamma)$. 
For the proof of Theorem~\ref{thm:main}, we show the following lemma involving 
$\Gamma_{m_{\lambda}}=\ch^{-1}(m_{\lambda})$ and binomial coefficients.
Let $\Gamma_{\gamma}(j)$ denote the class function  $\Gamma_{\gamma}(\cdot)$ evaluated at a 
permutation $\psi \in \SSS_n$ with cycle type $2^j,1^{n-2j}$.  For $0 \leq i \leq \nhalf$, define
\begin{equation}
\label{eqn:main}
  \alpha_i(\gamma) = \sum_{j=0}^i \binom{i}{j} \Gamma_{\gamma}(j). 
\end{equation}

In Section~\ref{sec:gmfs_monomial},
we prove the following lemma which we believe is of independent interest.

\begin{lemma}
\label{lem:frob_inv_monomial}
For all $\lambda \vdash n$ and for  $0\leq i \leq \nhalf$, 
%the quantity  
%$\alpha_{i}^{}(m_{\lambda})$ is a non-negative integral multiple of  $2^i$. 
%Moreover  for $0\leq i \leq \nhalf$,  
$\alpha_{i}^{}(m_{\lambda})  = 2^i$ if $\lambda=2^i,1^{n-2i}$ and 
$0$ otherwise.
\end{lemma}

In Section~\ref{sec:gmfs_forgotten_symm_fn}
we consider the generalized matrix polynomial of $\sL_T^q$ with 
respect to the {\it forgotten symmetric function}.  Our main result
of that section is Theorem~\ref{thm:main_f_lambda},
where we show similar monotonicity results as we move on the $\GTS_n$ poset.

\section{Preliminaries}
\label{sec:prelims}


We now give some motivational background for our results and place it in its context.  
The normalized immanant
of a matrix $A$ corresponding to a partition $\lambda$ is 
defined as $\displaystyle \od_{\lambda}(A) = \frac{d_{\lambda}(A)}{ \chi_{\lambda}^{}(\id)}$.
Here, $\chi_{\lambda}^{}(\id)$ equals the dimension of the irreducible representation
of $\SSS_n$ over $\CC$ indexed by $\lambda$.  


Schur \cite{schur-immanant-ineqs} showed 
that among normalized immanants, the determinant is the smallest 
normalized immanant for any positive semidefinite Hermitian matrix.
This shows that for any positive semidefinite Hermitian matrix $A$, all its
normalized immanants (and hence immanants) are non-negative.
Though the immanant $d_{\lambda}(A)$ is non-negative, \eqref{eqn:immanant}
is not a non-negative expression for $d_{\lambda}(A)$ as 
$\chi_{\lambda}^{}(\psi) \prod_{i=1}^n a_{i,\psi(i)}$ is not necessarily 
non-negative for all
$\psi \in \SSS_n$ that contribute to $d_{\lambda}(A)$.  

 Recall that the
Laplacian matrix $L_G$ of a graph $G$ is defined as $L_G = D-A$, 
where $D$ is the diagonal matrix with degrees of $G$ on the diagonal and $A$ 
is the adjacency matrix of $G$.  It is well known 
that $L_G$ is positive semidefinite for all graphs $G$ (see \cite{godsil-royle}).
When the matrix $A$ is the  Laplacian $L_T$ of a tree $T$,
then Chan and Lam in \cite{hook_immanant_explained-chan_lam} gave the
following two results which give an alternate positive expression for the immanant $d_{\lambda}(L_T)$.


We denote the number of parts of a partition $\lambda$ of $n$  as $l(\lambda)$.
For $\lambda \vdash n$, let $\chi_{\lambda}^{}(j)$ denote the character value 
$\chi_{\lambda}^{}(\cdot)$
evaluated at a permutation $\psi \in \SSS_n$ with cycle type $2^j, 1^{n-2j}$.
For $0 \leq i \leq \nhalf$ and $\lambda \vdash n$, define 
\begin{equation}
  \alpha_{i,\lambda} = \sum_{j=0}^i \binom{i}{j} \chi_{\lambda}^{}(j). \label{eqn:main_char}
\end{equation}




\begin{lemma}[\sc Chan and Lam, \cite{chan-lam-binom-coeffs-char}]
\label{lem:chan_lam_bino_char}
For all $\lambda \vdash n$ and for  $0\leq i \leq \nhalf$, the quantity  
$\alpha_{i, \lambda}$ is a non-negative integral multiple of $2^i$.  
Moreover  for $1\leq i \leq \nhalf$, 
$\alpha_{i,\lambda}=0$ if and only if  $l(\lambda)>n-i$. 
\end{lemma}


\begin{theorem}[\sc Chan and Lam, \cite{hook_immanant_explained-chan_lam}]
  \label{thm:positive_immanant}
 Let $L_T$ be the Laplacian matrix of a tree $T$ on $n$ vertices. Then,
 for all $\lambda \vdash n$, 
 $d_{\lambda}(L_T) = \sum_{i=0}^{\nhalf} \alpha_{i, \lambda} a_i(T)$,
 where $a_i(T)$ equals the number of vertex orientations with exactly 
 $i$  bidirected edges (and is hence a non-negative integer for all $i$).
\end{theorem}


Lemma~\ref{lem:chan_lam_bino_char} and Theorem~\ref{thm:positive_immanant} make it clear that all immanants 
of  $L_T$ are non-negative.  Similar results are known for the $q$-Laplacian of $T$  (see Theorem~\ref{thm:mukesh-siva-positive_immanant}).  Define 
$\sL_G^q = I+q^2(D-I) -qA$ as the $q$-Laplacian of a graph $G$,  where $D$ and $A$ are as before and $I$ is 
the identity matrix.  Here $q$ is a variable.   It is easy to see that
setting $q=1$ in $\sL_G^q$ gives us the usual combinatorial Laplacian $L_G$.
The matrix $\sL_G^q$ has appeared in the contexts
of Ihara--Selberg zeta functions of graphs $G$ (see Bass \cite{bass} and  Foata and Zeilberger \cite{foata-zeilberger-bass-trams}). When the graph $G$ 
is a tree $T$,  $\sL_T^q$ has connections with  
the inverse of the exponential distance matrix of $T$ (see Bapat, Lal and Pati 
\cite{bapat-lal-pati} and Nagar \cite{nagar}).  
Nagar and  Sivasubramanian in \cite{mukesh-siva-hook} gave the following 
$q$-analogue of Theorem~\ref{thm:positive_immanant} 
involving %hereby generalising Theorem 
%\ref{thm:positive_immanant}  to 
$\sL_T^q$. To state it, we need a 
$q$-analogue of the term $a_i(T)$.
We describe it briefly and refer the reader to 
\cite{mukesh-siva-hook} for more details.
Let $\sO_i(T)$ denote the set of vertex orientations in $T$ 
with $i$ bidirected edges, and let $a_i(T) = |\sO_i(T)|$.  A 
statistic $\law: \sO_i(T) \mapsto \ZZ_{\geq 0}$ was defined in 
\cite{mukesh-siva-hook}.   Using it, when $i \geq 1$, the following 
$q$-analogue $a_i(T,q)$ of $a_i(T)$ was defined 
(see \cite[Corollary~9]{mukesh-siva-hook}): 
\begin{equation}
\label{eqn:q-analog-vertex-orient}
a_i(T,q) = \sum_{O \in \sO_i} q^{\law(O)}.
\end{equation}
In \cite{mukesh-siva-hook}, we had also defined $a_0(T,q) = 1-q^2$.
It turns out that for all $i \geq 0$, the $a_i(T,q)$ is a polynomial 
in $q^2$ (see \cite[Remark~15]{mukesh-siva-hook}).
With this definition, we can state a $q$-analogue of Theorem~\ref{thm:positive_immanant}.


\begin{theorem}[\sc Nagar and Sivasubramanian]
  \label{thm:mukesh-siva-positive_immanant}
 Let $\sL_T^q$ be the $q$-Laplacian matrix of a tree $T$ on $n$ vertices. Then,
 for all $\lambda \vdash n$, 
 $d_{\lambda}(\sL_T^q) = \sum_{i=0}^{\nhalf} \alpha_{i, \lambda}  a_i(T,q)$,
 where the $a_i(T,q)$ is defined in \eqref{eqn:q-analog-vertex-orient}.
%is a polynomial in $q^2$, counting vertex 
% orientations with exactly  $i$  bidirected edges with respect to a
% statistic $\law(\cdot)$.
\end{theorem}

Clearly, setting $q=1$ in $\sL_T^q$ gives $L_T$ and 
setting $q=1$ in $a_i(T,q)$ gives $a_i(T)$ 
for $i$ with $0\leq i \leq \nhalf$.
%(see \cite{mukesh-siva-hook} for the  definitions).    
%Further, since $a_i(T,q)$ is a polynomial
%in $q^2$, Theorem   %\ref{thm:mukesh-siva-positive_immanant} holds for
%all $q \in \RR$.  
Extensions of Theorem~\ref{thm:mukesh-siva-positive_immanant}
to the bivariate $q,t$-Laplacian denoted 
as $\sL_T^{q,t}$ were also given in \cite{mukesh-siva-hook}. 
%and a counterpart of Theorem
%\ref{thm:mukesh-siva-positive_immanant} holds even when
%$q,t \in \RR$ with $qt > 0$ and when $q,t \in \CC$ with $qt > 0$.
Later, in \cite{mukesh-siva-gts}, the following more general result 
about the coefficient of $(-1)^rx^{n-r}$ in the immanantal 
polynomial $d_{\lambda}(xI-\sL_T^q)$  was proved.  To state it, we 
need polynomials $a_{i,r}^{}(T,q)$ which generalize
$a_i(T,q)$.  Let $B \subset V(T)$ be a subset of the vertex
set of $T$ with $|B| = r$, where $r \leq n$.  
%is a polynomial in $q^2$, 
Let $\sO_{B,i}^T$ be the set of vertex orientations of vertices
in $B$ that have $i$ bidirected edges.  There is a statistic 
$\awy_B^T: \sO_{B,i}^T \mapsto \ZZ_{\geq 0}$ with respect to which
we define
\begin{equation}
\label{eqn:defn_a_riT}
a_{i,r}^{}(T,q) = \sum_{B \subseteq V(T), |B|=r} 
\sum_{O \in \sO_{B,i}^T}q^{\awy(_B^T(O)}.
\end{equation}
With this definition, we have the following \cite[Lemma~7, Corollary~12]{mukesh-siva-gts}.
%which counts vertex orientations summed over vertex subsets $B$ having 
%exactly $r$ vertices in $T$ and with exactly $i$  bidirected 
%edges with respect to a statistic $\awy_B^T(\cdot)$.


\begin{lemma}[\sc Nagar and Sivasubramanian]
\label{lem:coeff_imm_poly}
Let $T$ be a tree on $n$ vertices with $q$-Laplacian $\sL_T^q$. 
For $\lambda \vdash n$, let 
$d_{\lambda}(xI-\sL_T^q)=\sum_{r=0}^n (-1)^r 
c_{\lambda,r}^{\sL_{T}^q}(q) x^{n-r}$. Then for $0\leq r \leq n$, we have 
$c_{\lambda,r}^{\sL_{T}^q}(q)
=\sum_{i=0}^{\rhalf}\alpha_{i,\lambda}^{} a_{i,r}^{}(T,q)$.
%where $a_{i,r}^{}(T,q)$ is a polynomial in $q^2$, counting vertex 
%orientations on all vertex subsets $B$ having exactly $r$ vertices 
%in $T$ and with exactly $i$  bidirected edges with respect to a
%statistic $\awy_B^T(\cdot)$.
\end{lemma}

In Lemma~\ref{lem:coeff_imm_poly}, 
one can get the immanant $d_{\lambda}(\sL_q^T)$  as the 
constant term of the immanantal polynomial (that is, by 
looking at the special case when $r=n$).
%in $ a_{i,r}^{}(T,q)$ clearly 
%gives $ a_{i}^{}(T,q)$  for all $i$ (see \cite{mukesh-siva-gts} 
%for the  definition of $ a_{i,r}^{}(T,q)$). 
Thus, Lemma~\ref{lem:coeff_imm_poly} generalizes  
Theorem~\ref{thm:mukesh-siva-positive_immanant}. 
Using Lemma~\ref{lem:coeff_imm_poly}, the following
result  about monotonicity for the coefficients of all
immanantal polynomials of $\sL_{T}^q$ along $\GTS_n$ was proved in \cite[Theorem~1]{mukesh-siva-gts}.

\begin{theorem}[\sc Nagar and Sivasubramanian]
\label{thm:main_earlier}
Let $T_1$ and $T_2$ be two trees with $n$ vertices such that $T_2$ covers $T_1$ 
in $\GTS_n$.  Let $\sL_{T_1}^q$ and $\sL_{T_2}^q$ be the $q$-Laplacians 
of $T_1$ and $T_2$ respectively.  For $\lambda \vdash n$, let 
\begin{align*}
  \zeta^{\sL_{T_1}^q}_{s_{\lambda}}(x) & =  d_{\lambda}(xI -\sL_{T_1}^q) = \sum_{r=0}^n (-1)^r 
c_{\lambda,r}^{\sL_{T_1}^q}(q) x^{n-r} \mbox{ and} \\
\zeta^{\sL_{T_2}^q}_{s_{\lambda}}(x) & =  d_{\lambda}(xI -\sL_{T_2}^q) = \sum_{r=0}^n (-1)^r 
c_{\lambda,r}^{\sL_{T_2}^q}(q) x^{n-r}.
\end{align*}
Then for all $\lambda \vdash n$, we have 
$c_{\lambda,r}^{\sL_{T_1}^q}(q) - c_{\lambda,r}^{\sL_{T_2}^q}(q) 
\in \RR^+[q^2]$, where $0 \leq r \leq n$.   
\end{theorem}

The proof of Theorem~\ref{thm:main_earlier} required to give a combinatorial expression for 
the coefficients $c_{\lambda,r}^{\sL_{T_1}^q}(q)$ and $c_{\lambda,r}^{\sL_{T_2}^q}(q)$ (see Lemma~\ref{lem:coeff_imm_poly}) and then to give
an injection between the objects counting $c_{\lambda,r}^{\sL_{T_2}^q}(q)$
and those counting $c_{\lambda,r}^{\sL_{T_1}^q}(q)$ (see Lemma~\ref{lem:diff_a_k,r(T,q)}).







\section{GMF of $q$-Laplacians arising from symmetric functions}
\label{sec:gmfs_symm_fns}


Let $\Gamma_{\gamma}=\ch^{-1}(\gamma)$ be the inverse Frobenius
image of the symmetric function $\gamma \in \Lambda_{\QQ}^n$. 
Note that $\Gamma_{\gamma}$ is a class function of $\SSS_n$ over $\CC$ indexed by $\gamma$. 
Plugging the matrix $\sL_T^q=(l_{i,j}^q)_{1\leq i,j \leq n }$ 
in \eqref{eqn:gen_matrix_fn}, we get 
\begin{equation}
  \label{eqn:gen_matrix_fn_sL}
  d_{\gamma}(\sL_T^q) = \sum_{\psi \in \SSS_n} \Gamma_{\gamma}(\psi) \prod_{i=1}^n l_{i,\psi(i)}^q.
\end{equation}

%Recall the polynomial $a_i(T,q)$ that appears in Theorem
%\ref{thm:mukesh-siva-positive_immanant} and that 
For the matrix $\sL_T^q$, it is very easy to show the following
counterpart of Theorem~\ref{thm:mukesh-siva-positive_immanant}, 
where we change the Frobenius inverse of the Schur symmetric
function to the Frobenius inverse of an arbitrary symmetric function.
We recall the polynomial $a_i(T,q)$ from \eqref{eqn:q-analog-vertex-orient}
%Lemma \ref{lem:coeff_imm_poly} 
and $\alpha_i(\gamma)$ from \eqref{eqn:main}.  We start with the 
following result.  As its proof is a verbatim copy of the 
proof of Theorem~\ref{thm:mukesh-siva-positive_immanant}, with 
the only change being that we replace 
$\alpha_{i,\lambda}$ by $\alpha_i(\gamma)$, we omit it. 


\begin{theorem}
  \label{thm:mukesh-siva-gen_matrix_fn}
 Let $\sL_T^q$ be the $q$-Laplacian matrix of a tree $T$ on $n$ vertices. 
 Then, for all $\gamma \in \Lambda_{\QQ}^n$, 
 $d_{\gamma}(\sL_T^q) = \sum_{i=0}^{\nhalf} \alpha_i (\gamma) a_i(T,q)$.
\end{theorem}

It is very simple to show the following counterpart of Lemma~\ref{lem:coeff_imm_poly}.  Recall the polynomial $a_{i,r}(T,q)$ from
\eqref{eqn:defn_a_riT}.  Since
the proof is identical, we omit it and merely state the result.

\begin{lemma}
	\label{lem:coeff_gmf_poly}
	Let $T$ be a tree on $n$ vertices with $q$-Laplacian $\sL_T^q$. 
	Then for all $\gamma \in \Lambda_{\QQ}^n$, we have 
	$d_{\gamma}(xI-\sL_T^q)=\sum_{r=0}^n (-1)^r 
	c_{\gamma,r}^{\sL_{T}^q}(q) x^{n-r}$, where 
	$c_{\gamma,r}^{\sL_{T}^q}(q)
	=\sum_{i=0}^{\rhalf}\alpha_{i}(\gamma)a_{i,r}^{}(T,q)$ for $0 \leq r \leq n$.
\end{lemma}

Every element $\gamma \in \Lambda_{\QQ}^n$ is written as a linear 
combination of basis vectors of $\Lambda_{\QQ}^n$.
The vector space $\Lambda_{\QQ}^n$ has six standard bases.
We use standard terminology to 
denote each of the usual bases of $\Lambda_{\QQ}^n$.  Thus,
$s_{\lambda}$, $p_{\lambda}$, $e_{\lambda}$, $h_{\lambda}$, 
$m_{\lambda}$ and $f_{\lambda}$  denote the
Schur,  power sum, elementary, homogeneous, monomial 
and forgotten symmetric functions respectively. 
The inverse Frobenius map of each of these will be denoted by the
same letter, but in capital font and with the partition $\lambda$ 
as a superscript
rather than a subscript.  Thus, $E^{\lambda} = \ch^{-1}(e_{\lambda})$,
$\chi_{\lambda}^{} = S^{\lambda} = \ch^{-1}(s_{\lambda})$ and so on. 

As mentioned earlier, the inverse Frobenius
image $\ch^{-1}(s_{\lambda})$ of the Schur symmetric function $s_{\lambda}$ 
is $\chi_{\lambda}^{}$,
the irreducible character of $\SSS_n$ over $\CC$ indexed by $\lambda$. Therefore the identity given in \eqref{eqn:main_char} is a special case 
of the identity given in \eqref{eqn:main} when $\gamma=s_{\lambda}$.
Thus, if any symmetric function $\gamma \in \Lambda_{\QQ}^n$ is Schur-positive
(that is, $\gamma = \sum_{\lambda \vdash n} a_{\lambda} s_{\lambda}$ where
$a_{\lambda} \in \RR^{+}$ for all $\lambda \vdash n$), 
then, by linearity, Lemma~\ref{lem:chan_lam_bino_char} 
will be true with $\chi_{\lambda}^{}$ replaced by $\ch^{-1}(\gamma)$ in \eqref{eqn:main_char}.   
Unfortunately, the monomial symmetric function $m_{\lambda}$ 
indexed by $\lambda$ is not Schur-positive.
Thus, if $M^{\lambda} = \ch^{-1}(m_{\lambda})$, it is not clear
that Lemma \ref{lem:chan_lam_bino_char} 
with $\chi_{\lambda}^{}$ replaced by $M^{\lambda}$ 
in \eqref{eqn:main_char} is true.  





For $\gamma \in \Lambda_{\QQ}^n$, let $\Gamma_{\gamma} = \ch^{-1}(\gamma)$. 
%In $\alpha_i^{}(\gamma)$, since we only need to evaluate  $\Gamma_{\gamma}(\psi)=\Gamma_{\gamma}(j)$ for 
%a permutation $\psi \in \SSS_n$ with cycle type $2^j, 1^{n-2j}$.  
Thus, if $\Gamma_{\gamma}(j) \geq 0$
for all $j$, then from \eqref{eqn:main}, we get 
$\alpha_i(\gamma) \geq 0$.   Since the inverse Frobenius image $\Gamma_{p_{\lambda}}$ of
$p_{\lambda}$ is a scalar multiple of the indicator function of 
the conjugacy class $C_{\lambda}$ indexed by $\lambda$ (see \cite{EC2}), it follows
that $\alpha_i^{}(p_{\lambda}) \geq 0$ for all $\lambda \vdash n$ and for all $i=0,1,\ldots,\nhalf$.

As mentioned earlier, if $\gamma \in \Lambda_{\QQ}^n$
is Schur-positive, then $\alpha_i^{}(\gamma) \geq 0$.  Since $h_{\lambda}$
is Schur-positive (see \cite[Corollary~7.12.4]{EC2}), it follows from
Lemma~\ref{lem:chan_lam_bino_char} that 
$\alpha_i^{}(h_{\lambda}) \geq 0$ for all $\lambda \vdash n$.  
Similarly, it is well known
that $e_{\lambda}$ is also Schur-positive.  Thus 
$\alpha_i^{}(e_{\lambda}) \geq 0$ for all $\lambda \vdash n$.  
We record these facts below for future use.

\begin{lemma}
  \label{lem:p_h_e_sum}
For all $\lambda \vdash n$ and for $0 \leq i \leq \nhalf$, we have 
$\alpha_i^{}(\gamma) \geq 0$ for 
$\gamma \in \{p_{\lambda}, h_{\lambda}, e_{\lambda}  \}.$
\end{lemma}

Let the tree $T$ have $n$ vertices and, for $0\leq i,r\leq n$, 
let $a_{i,r}^{}(T,q)=a_{i,r}^T(q)$ be the polynomial in $q^2$ with non-negative real coefficients, 
defined by Nagar and Sivasubramanian in \cite[page~7]{mukesh-siva-gts}.
They showed the following result  (see 
\cite[Lemma~19, 23 and Corollary~22]{mukesh-siva-gts}). 

\begin{lemma}[\sc Nagar and Sivasubramanian]
	\label{lem:diff_a_k,r(T,q)}
	Let $T_1$ and $T_2$ be two trees with $n$ vertices such that  $T_2$ covers $T_1$ in $\GTS_n$. 
	Then for all $0\leq i,r\leq n$, we have  $a_{i,r}^{}(T_1,q)-a_{i,r}^{}(T_2,q) \in \RR^+[q^2]$. 
\end{lemma}
With Lemma~\ref{lem:p_h_e_sum} and Lemma~\ref{lem:diff_a_k,r(T,q)}
in place, we can get the following Corollary.  Its 
proof mimics the proof of Theorem~\ref{thm:main_earlier}.



\begin{corollary}
  \label{cor:three}
Theorem~\ref{thm:main_earlier} is true  when we replace the 
immanantal polynomial $d_{\lambda}(xI - \sL_{T_j}^q)$ by 
the generalized matrix polynomials   
$d_{\gamma}(xI - \sL_{T_j}^q)$ for 
$\gamma \in \{p_{\lambda}, h_{\lambda}, e_{\lambda}  \}$ and 
for $j=1,2$.
\end{corollary}


By Theorem~\ref{thm:main_earlier} and Corollary~\ref{cor:three}, 
we thus have monotonicity results about four of the six standard bases,  
as we move up on $\GTS_n$.
This paper plugs the gaps left by considering the
generalized matrix polynomials of $\sL_T^q$ arising from the last two 
bases $m_{\lambda}$ and $f_{\lambda}$. In other words, 
we consider the cases when we replace the 
immanantal polynomial $d_{\lambda}(xI - \sL_{T_j}^q)$ by 
$d_{g}(xI - \sL_{T_j}^q)$ when 
$g \in \{m_{\lambda}, f_{\lambda} \}$  
in Theorem~\ref{thm:main_earlier}.


\section{GMF of tree Laplacians arising from $m_{\lambda}$, the monomial symmetric function}
\label{sec:gmfs_monomial}

We prove Theorem~\ref{thm:main} in this section.  
We need the notion of 
$\lambda$-brick tabloids of shape $\mu$ which is used to give a combinatorial interpretation of 
the quantity $\Gamma_{m_{\lambda}}(\psi)=M^{\lambda}(\psi)$, where 
$\lambda, \mu\vdash n$ and $\psi\in\SSS_n.$ 
We recall the following definition of 
$\lambda$-brick tabloid of shape $\mu$ defined by  E\u{g}ecio\u{g}lu  
and Remmel in \cite{remmel-monomial-sym-frob}. 

\begin{definition}
Let 
$\lambda \vdash n$ with $\lambda =\lambda_1\geq \lambda_2\geq\cdots\geq \lambda_{l(\lambda)}^{}$. 
Let $\mu \vdash n$  
and let $F_{\mu}$ be its Ferrers diagram. 
A $\lambda$-brick tabloid  $\sB_{\lambda,\mu}$ of shape $\mu$ is a filling of
$F_{\mu}$ with bricks $b_1,b_2,\ldots,b_{l(\lambda)}^{}$ 
of size $\lambda_1, \lambda_2,\ldots, \lambda_{l(\lambda)}^{}$ respectively 
such that brick $b_i$ covers exactly $\lambda_i$ squares of $F_{\mu}$ 
all of which lie in a single row of $F_{\mu}$ and no two bricks overlap. 
Here, bricks of the same size are indistinguishable. 	
\end{definition}


We refer the reader to the book by Mendes and Remmel  \cite[Chapter~2]{mendes-remmel-book} for
an introduction to $\lambda$-brick tabloids. 
Given a brick $b$ in $\sB_{\lambda,\mu}$, let $|b|$ denote the length of $b$. 
Define  $\wt_{\sB_{\lambda,\mu}}^{}(b)$ to be $|b|$ 
if $b$ is at the end of a row in $\sB_{\lambda,\mu}$ and $1$ otherwise. 
We next define a weight $w(\sB_{\lambda,\mu})$ for each 
$\lambda$-brick tabloid $\sB_{\lambda,\mu}$ of shape $\mu$ as follows: 
\begin{equation*}
\label{eqn:def_wt_bricks}
w(\sB_{\lambda,\mu})=\prod_{b\in \sB_{\lambda,\mu}} \wt_{\sB_{\lambda,\mu}}^{}(b).
\end{equation*}

In other words, $w(\sB_{\lambda,\mu})$ is the product of the 
lengths of the rightmost brick in each row of $\sB_{\lambda,\mu}$.
For $\lambda, \mu \vdash n$,  
let $\BT_{\lambda,\mu}$ be the set of all $\lambda$-brick tabloids of shape $\mu$. 
For $\lambda \vdash n$ and for $\psi \in \SSS_n$, 
 E\u{g}ecio\u{g}lu  and Remmel in 
\cite[Theorem~1]{remmel-monomial-sym-frob} gave the following 
combinatorial interpretation of $M^{\lambda}(\psi)$
which we need. 

\begin{theorem}[\sc E\u{g}ecio\u{g}lu  and Remmel]
	\label{thm:main_thm_remmel_egec}
	For $\lambda \vdash n$, let  $M^{\lambda}=\ch^{-1}(m_{\lambda})$. 
	When a permutation $\psi \in \SSS_n$ has cycle type $\mu$, then 
	$$M^{\lambda}(\psi)=(-1)^{l(\lambda)-l(\mu)}\sum_{\sB_{\lambda,\mu} 
\in \BT_{\lambda,\mu}}w(\sB_{\lambda,\mu}).$$
\end{theorem}








We say $\lambda$ is a refinement of $\mu$ 
if the parts $\mu_i$ of $\mu$ can be obtained as a disjoint union of the 
parts $\lambda_j$ of $\lambda$. 
From the definition of $\sB_{\lambda,\mu}$, it is easy to see that 
if $\lambda$ is not a refinement of $\mu$,  then $\BT_{\lambda,\mu}=\emptyset$. 
Thus $M^{\lambda}(\psi)=0$, unless when $\lambda$ refines the cycle type $\mu$ of $\psi$. 
Let $M^{\lambda}(j)$ denote the value of the function $M^{\lambda}(\cdot)$ 
evaluated at a permutation $\psi \in \SSS_n$ with cycle type $2^j,1^{n-2j}$.  
We start with the following simple consequence of Theorem~\ref{thm:main_thm_remmel_egec}, 

\begin{corollary}
	\label{cor:remmel_mono}
	Let $\lambda \vdash n$ and let $\psi \in \SSS_n$ be a permutation 
with cycle type $\mu=2^j,1^{n-2j}$. Then, 
	$$M^{\lambda}(j)= 
	\begin{cases}
	(-1)^{j-k}2^k\binom{j}{k}, & \mbox{if } \lambda=2^k,1^{n-2k} \mbox{ and } k\leq j,  \\
	0, &  \mbox{otherwise.}
	\end{cases}
	$$
\end{corollary}

\begin{proof}
We divide the proof into two cases when $\lambda$ is  a 
refinement of $\mu$ and when it is not. 
If $\lambda$ is a refinement of $\mu$, then we must have  
$\lambda=2^k,1^{n-2k}$ for some 
$k$ with $0 \leq k\leq j.$ In this case, all the $k$ bricks of 
length $2$ must be  placed in some of the $k$ rows from the first $j$ 
rows of $F_{\mu}$. Thus, the number of 
such $\lambda$-brick tabloids $\sB_{\lambda,\mu}$ of 
shape $\mu$ is $\binom{j}{k}$, and each $\sB_{\lambda,\mu}$  
contributes the weight $2^k$ in the summation of   Theorem~\ref{thm:main_thm_remmel_egec}.
Thus,  the total contribution in  $M^{\lambda}(j)$ is  $2^k\binom{j}{k}$ and   $l(\lambda)-l(\mu)=j-k$. Hence  
$M^{\lambda}(j)= (-1)^{j-k}2^k\binom{j}{k}$.

When $\lambda$ is not a refinement of $\mu$, by 
Theorem~\ref{thm:main_thm_remmel_egec},  
$M^{\lambda}(j)=0$,  completing the proof. 
\end{proof}

%\vspace{2mm} 
We next prove Lemma~\ref{lem:frob_inv_monomial} which says that for all $\lambda \vdash n$ 
and for $0\leq i \leq \nhalf$, the quantity 
$\alpha_{i}^{}(m_{\lambda})$ is a non-negative integral  multiple of $2^i$. 
We will need the following very easy identity  involving
binomial coefficients
 $\binom{j}{k}\binom{i}{j}=\binom{i-k}{j-k}\binom{i}{k}$. 


\vspace{2mm}

\begin{proof}[Proof of Lemma 2]
By Corollary~\ref{cor:remmel_mono},  when $\lambda\neq 2^k,1^{n-2k}$, 
$M^{\lambda}(j)= 0$ for all $j$. Thus, from \eqref{eqn:main}, 
$\alpha_{i}^{}(m_{\lambda})=0$ for $ 0 \leq i \leq \nhalf.$ 
When $\lambda= 2^k,1^{n-2k}\vdash n$ for some $k$ 
with $0\leq k \leq \nhalf$, by Corollary~\ref{cor:remmel_mono}, we see that 
\begin{align*}
		\alpha_{i}^{}(m_{\lambda})  &=  \sum_{j=0}^{i}M^{\lambda}(j)\binom{i}{j} 
		=  
		\sum\limits_{j=0}^{i}(-1)^{j-k}2^k\binom{j}{k}\binom{i}{j}   \\
		 &= 
		2^k \binom{i}{k}\sum\limits_{j=k}^{i}(-1)^{j-k}\binom{i-k}{j-k} 
		=  
		\begin{cases}
			2^i, & \mbox{if } i=k,  \\
			0, &  \mbox{otherwise.}
		\end{cases} 
	\end{align*} 
	
	The proof is complete.
\end{proof}

\vspace{2mm}





Using Lemmas~\ref{lem:frob_inv_monomial} and  \ref{lem:diff_a_k,r(T,q)}, we can now prove Theorem~\ref{thm:main}.

\vspace{2mm}
\begin{proof}[Proof of Theorem 1]
From Lemmas~\ref{lem:frob_inv_monomial} and  \ref{lem:coeff_gmf_poly}, for $0 \leq r \leq n$ and for $j=1,2$, 
it is simple to  see that the coefficient of $(-1)^rx^{n-r}$ in $d_{m_{\lambda}}(xI-\sL_{T_j}^q)$ is given by 
\begin{equation}
\label{eqn:coeff_m}
c_{m_{\lambda},r}^{\sL_{T_j}^q}(q) 
=\sum_{i=0}^{\rhalf}\alpha_{i}(m_{\lambda})a_{i,r}^{}(T_j,q)
= 
\begin{cases}
2^k a_{k,r}^{}(T_j,q),& \mbox{if }  \lambda=2^k,1^{n-2k} \mbox{ and } k\leq \rhalf, \\
0, &  \mbox{otherwise.}
\end{cases} 
\end{equation}


By Lemma~\ref{lem:diff_a_k,r(T,q)} and \eqref{eqn:coeff_m}, we get
$c_{m_{\lambda},r}^{\sL_{T_1}^q}(q)-c_{m_{\lambda},r}^{\sL_{T_2}^q}(q)\in \RR^+[q^2]$ for all $\lambda\vdash n$ 
and for $0 \leq r \leq n.$ 
Furthermore, if  $\lambda\neq 2^k,1^{n-2k} \vdash n$ for all  $k$, then  from \eqref{eqn:coeff_m}, 
the  coefficient of $(-1)^rx^{n-r}$ in $d_{m_{\lambda}}(xI-\sL_{T_j}^q)$ 
is zero for $0 \leq r \leq n$ and $j=1,2$. 
The proof is complete.
\end{proof}

\vspace{2mm}

Thus, for all $\lambda\vdash n$ and for all $q\in \RR$,   moving up on $\GTS_n$ decreases the 
coefficient of $(-1)^rx^{n-r}$ in $d_{m_{\lambda}}(xI-\sL_{T}^q)$, 
where $0 \leq r \leq n$. 
Consequently, for all positive integers $n$,  this monotonicity result on $\GTS_n$ shows that the 
max-min pair of these coefficients is  $(P_n,S_n)$, where $P_n$ 
and $S_n$ are the path tree and the star tree on $n$ vertices respectively.
The following example illustrates Lemma~\ref{lem:frob_inv_monomial}.

\begin{example} 
	Let $n=15$. For all $\lambda\vdash n$ and for all $i\in \{0,1,2,3,4,5,6,7\}$, 
the values of  the quantity $\alpha_i(m_{\lambda})$ are tabulated in Table \ref{tab:alpha_r_m_lambda}. 
	
\begin{table}[h]
	$$ 	\begin{array}{|l||c|c|c|c|c|c|c|c|}
	\hline
	\lambda \vdash n=15 & i=0 & i=1 & i=2 & i=3  & i=4 & i=5 & i=6 & i=7	 \\
	\hline 
	\hline 
	\lambda= 1^{15} & 1 &	0 & 0 & 0 & 0 & 0 & 0 & 0\\
	\hline
	\lambda= 2,1^{13} & 0 &	2 & 0 & 0 & 0 & 0 & 0 & 0\\
	\hline
	\lambda= 2^2,1^{11} & 0 &	0 & 4 & 0 & 0 & 0 & 0 & 0\\
	\hline
	\lambda= 2^3,1^{9} & 0 &	0 & 0 & 8 & 0 & 0 & 0 & 0\\
	\hline
	\lambda= 2^4,1^{7} & 0 &	0 & 0 & 0 & 16 & 0 & 0 & 0\\
	\hline
	\lambda= 2^5,1^{5} & 0 &	0 & 0 & 0 & 0 & 32 & 0 & 0\\
	\hline
	\lambda= 2^6,1^{3} & 0 &	0 & 0 & 0 & 0 & 0 & 64 & 0\\
	\hline
	\lambda= 2^7,1 & 0 &	0 & 0 & 0 & 0 & 0 & 0 & 128\\
	\hline
	\lambda\neq 2^k,1^{n-2k} & 0 &	0 & 0 & 0 & 0 & 0 & 0 & 0\\
	\hline
	\end{array} $$
	\caption{The value of $\alpha_i(m_{\lambda})$.}
		\label{tab:alpha_r_m_lambda}
\end{table}

\end{example}


\section{GMF of tree Laplacians arising from $f_{\lambda}$, the forgotten symmetric function}
\label{sec:gmfs_forgotten_symm_fn}

Let $f_{\lambda}$ denote the forgotten 
symmetric function indexed by  $\lambda \vdash n$ and let 
$F^{\lambda}=\ch^{-1}(f_{\lambda})$ denote its inverse Frobenius 
image. 
This section is devoted to show monotonicity of the coefficient of $(-1)^rx^{n-r}$ in $\zeta_{f_{\lambda}}^{\sL_T^q}(x)=d_{f_{\lambda}}(xI-\sL_T^q)$ when we move up along $\GTS_n$.  
For this, we need the following combinatorial interpretation of $F^{\lambda}(\psi)$ by
E\u{g}ecio\u{g}lu  and Remmel in  \cite[Theorem~8]{remmel-monomial-sym-frob}.  
 
 \begin{theorem}[\sc E\u{g}ecio\u{g}lu  and Remmel]
 	\label{thm:remmel_forgottan_inte}
 Let $\lambda \vdash n$ and let $\psi \in \SSS_n$ be a permutation with cycle type $\mu$. Then, 
 $$F^{\lambda}(\psi)=(-1)^{n-l(\mu)}\sum_{\sB_{\lambda,\mu} \in \BT_{\lambda,\mu}}w(\sB_{\lambda,\mu}).$$
 	\end{theorem}
 
 For $\lambda \vdash n$, let  $F^{\lambda}(j)$ denote the value of the function  
$F^{\lambda}(\cdot)$ evaluated at a permutation $\psi \in \SSS_n$ with cycle type $2^j,1^{n-2j}$. 
By Theorem~\ref{thm:remmel_forgottan_inte}, the proof of the following 
corollary is 
identical to the proof of Corollary~\ref{cor:remmel_mono}, we omit it and merely state the result.
\begin{corollary}
	\label{cor:remmel_forgottan}
	For $\lambda \vdash n$ and for  $0\leq j \leq \nhalf$,  we have 
	$$F^{\lambda}(j)= 
	\begin{cases}
	(-1)^{j}2^k\binom{j}{k}, & \mbox{if } \lambda=2^k,1^{n-2k} \mbox{ and } k\leq j,  \\
	0, &  \mbox{otherwise.}
	\end{cases} 
	$$
\end{corollary} 

For a fixed $\lambda \vdash n$, let $\gamma=f_{\lambda}$ in \eqref{eqn:main}. 
We next calculate the quantity $\alpha_{i}(f_{\lambda})$ in the following lemma 
which will be used later in the determination of the coefficient of $(-1)^rx^{n-r}$ in $d_{f_{\lambda}}(xI-\sL_{T}^q).$ 
\begin{lemma}
	\label{lem:alpha_f}
	Let $\lambda \vdash n$. 
	Then for $0\leq i \leq \nhalf$,  we have 
	$$\alpha_{i}^{}(f_{\lambda})= 
	\begin{cases}
	(-1)^i 2^i, & \mbox{if } \lambda=2^i,1^{n-2i},  \\
	0, &  \mbox{otherwise.}
	\end{cases} 
	$$
\end{lemma}

\begin{proof}
By Corollary~\ref{cor:remmel_forgottan} when $\lambda\neq 2^k,1^{n-2k}$,  
$F^{\lambda}(j)=0$ for all $j$. By \eqref{eqn:main},  $\alpha_{i}^{}(f_{\lambda})=0$ for $0 \leq i \leq \nhalf.$ 
We next assume $\lambda= 2^k,1^{n-2k}\vdash n$ for some $k$ 
with $0\leq k \leq \nhalf$. In this case, by Corollary~\ref{cor:remmel_forgottan}, we see that 
\begin{align*}
\alpha_{i}^{}(f_{\lambda}) & =  \sum_{j=0}^{i}F^{\lambda}(j)\binom{i}{j} 
=  
\sum\limits_{j=0}^{i}(-1)^{j}2^k\binom{j}{k}\binom{i}{j}   \\
 & =  
2^k \binom{i}{k}\sum\limits_{j=k}^{i}(-1)^{j}\binom{i-k}{j-k} 
			% = 
%	(-1)^k	2^k \binom{i}{k}\sum\limits_{i=0}^{i}(-1)^{i}\binom{i-k}{i} \\&	
=		\begin{cases}
			(-1)^i	2^i, & \mbox{if } i=k,  \\
			0, &  \mbox{otherwise.}
		\end{cases} 
	\end{align*}
	All equalities above follow by simple 
	manipulations and hence the proof is complete.
\end{proof}

\begin{theorem}
	\label{thm:main_f_lambda}
Let $T$ be a tree on $n$ vertices with $q$-Laplacian matrix $\sL_{T}^q$. 
Then  for $0\leq r \leq n$, the coefficient of $(-1)^rx^{n-r}$ in $d_{f_{\lambda}}(xI-\sL_{T}^q)$ is given by 
\begin{equation*}
\label{eqn:coeff_f}
c_{f_{\lambda},r}^{\sL_{T}^q}(q)
=
\begin{cases}
(-1)^k2^k a_{k,r}(T,q), & \mbox{if } \lambda=2^k,1^{n-2k} \mbox{ with } k \leq \rhalf, \\
0, &  \mbox{otherwise.}
\end{cases}
\end{equation*} 
Furthermore, for all $\lambda\vdash n$
and all $q\in \RR$,
moving up on $\GTS_n$ 
decreases $c_{f_{\lambda},r}^{\sL_{T}^q}(q)$ in absolute value.
\end{theorem} 

\begin{proof}
	From Lemmas~\ref{lem:coeff_gmf_poly} and  \ref{lem:alpha_f},  
it is simple to see that 	
	\begin{equation*}
	c_{f_{\lambda},r}^{\sL_{T}^q}(q) 
	=\sum_{i=0}^{\rhalf}\alpha_{i}(f_{\lambda})a_{i,r}^{}(T,q)
	= 
	\begin{cases}
	(-1)^k2^k a_{k,r}^{}(T,q),& \mbox{if }  \lambda=2^k,1^{n-2k} \mbox{ and } k\leq \rhalf, \\
	0, &  \mbox{otherwise.}
	\end{cases} 
	\end{equation*}
	Let $T_1$ and $T_2$ be two trees with $n$ vertices such that  $T_2$ covers $T_1$ in $\GTS_n$.  
By Lemma~\ref{lem:diff_a_k,r(T,q)},  
$\left|c_{f_{\lambda},r}^{\sL_{T_1}^q}(q)\right|-\left|c_{f_{\lambda},r}^{\sL_{T_2}^q}(q)\right|\in \RR^+[q^2]$ 
completing the proof.
\end{proof}

\vspace{2mm}
Thus, for all $\lambda\vdash n$ when $ 0 \leq r \leq n$, 
by Theorem~\ref{thm:main},  Theorem~\ref{thm:main_f_lambda} and Corollary~\ref{cor:three}, we get 
that the coefficient of $x^r$ in $d_{\gamma}(xI-\sL_{T}^q)$ 
decreases as we move up along $\GTS_n$ in absolute value for each  
$\gamma\in \{m_{\lambda},s_{\lambda},p_{\lambda},h_{\lambda},e_{\lambda},f_{\lambda}\}$.  Plugging in
 $q=1$, we get the following corollary of 
Theorem~\ref{thm:main}, Theorem~\ref{thm:main_f_lambda} and Corollary~\ref{cor:three}. 

\begin{corollary}
For all $\lambda\vdash n$ and for all 
$\gamma\in \{m_{\lambda},s_{\lambda},p_{\lambda},h_{\lambda},e_{\lambda},f_{\lambda}\}$, 
moving up on $\GTS_n$ decreases  the  coefficient of $x^r$ in $d_{\gamma}(xI-L_T)$ in absolute value, 
for $0 \leq r \leq n$.  Here $L_T$ is the usual combinatorial Laplacian of the tree $T$.
\end{corollary}

\section*{Acknowledgement}
The first author would like to 
acknowledge SERB, Government of India for providing a national postdoctoral
fellowship with the file number PDF/2018/000828.  
The second author acknowledges support
from project MTR/2018/000453 
given by SERB, Government of India.

Our main theorem in this work was in its conjecture form, tested
using the computer package ``SageMath''. We thank the authors for generously releasing SageMath as an open-source package.

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\end{document}

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