    \section{Introduction}

Chip-firing is a game that takes place on a connected graph $G$, where chips are distributed across the vertices of $G$ and moved to adjacent vertices based on a straightforward rule. This dynamical system has a profound theory that links to various fields in mathematics and physics \cite{BakTangWiesenfeld88,Dhar90,Gabrielov93,digraph}. For further details, please see the recent textbooks \cite{Klivans} and \cite{CorryPerk}.

Consider a simple graph $G$ with $n + 1$ vertices, where one vertex is designated as the \emph{sink}, and chips are placed on each non-sink vertex. The allocation of chips is described by an integer vector $\vec{c} \in \mathbb{Z}^{n}$, known as a \textit{(chip) configuration}. A non-sink vertex with chips equal to or greater than its degree can fire, distributing one chip along each incident edge to adjacent vertices. A configuration $\vec{c}$ is termed \emph{stable} if no non-sink vertex is able to fire.

For a connected graph $G$, any starting configuration will eventually reach stability via sequence of firings, with some chips moving to the sink vertex. Letting $v_1, ..., v_{n}$ represent the non-sink vertices of $G$, the chip-firing rules can be described using $L_G$, the $n \times n$ \textit{reduced Laplacian} matrix of $G$.

The outcome of firing a vertex $v_i$ on a chip configuration $\vec{c}$ is represented by $\vec{c} - L\vec{e_i}$, where $\vec{e_i}$ denotes the $i$-th standard basis vector. The matrix $L_G$ establishes an equivalence relation among the vectors in ${\mathbb Z}^{n}$, with $\vec{c}$ and $\vec{d}$ being \emph{firing equivalent} if $\vec{c} - \vec{d}$ lies within the image of $L_G$. This determines the \emph{critical group} of $G$, as described by $\mathcal{K}(G) := {\mathbb Z}^n/\Img L_G$ \cite{Biggs99}.

A configuration $\vec{c}$ is considered \emph{valid} (or \emph{effective}) if for all $i = 1, \dots, n$, the condition $c_i \geq 0$ holds. The goal is to identify notable valid configurations within each equivalence class $[\vec{c}] \in {\mathcal K}(G)$. It turns out that each $[\vec{c}]$ contains a distinct valid configuration that is \emph{critical}, which means it is stable and can be derived by iteratively firing from a sufficiently large configuration $\vec{b}$. Stabilizing the sum of two critical configurations results in a critical configuration.

One can show that each $[\vec{c}]$ contains a unique valid configuration that is \emph{superstable}, which means that it is stable under \emph{set-firings}, the simultaneous firing of a set of vertices.  A superstable configuration represents a solution to an energy minimization problem and aligns with the concept of a \emph{$G$-parking function}. 

For a connected graph $G$, there is a straightforward bijection linking the set of critical configurations and the set of superstable configurations, which are both correspondingly in bijection with the set of spanning trees of $G$. This simple map (taking a configuration, negating it coordinate-wise from a certain maximal configuration) is what is called the \newword{duality map} between superstable configurations and critical configurations, and our focus is to extend this map to more general models.

Recently, chip-firing has been extended to more general settings, where the reduced Laplacian of a graph is replaced by other matrices (see, for instance, \cite{gabrielov,GuzKliMmatrices}). We use a matrix $M$ to define a firing rule that mimics the graphical setting: firing $v_i$ now takes a configuration $\vec{c}$ to $\vec{c} - M \vec{e_i}$, where $\vec{e_i}$ stands for the unit vector with $1$ at the $i$-th coordinate. For a well-defined notion of chip-firing we require that $M$ satisfies an \emph{avalanche finite} property, so that repeated firings of any initial configuration eventually stabilize in an appropriate sense.  The class of matrices with this property are known as \emph{$M$-matrices}, and can be characterized in a number of ways (see \Cref{def:Mmatrix} below). In \cite{GuzKliMmatrices}, Guzm\'an and Klivans have shown that the chip-firing theory defined by an $M$-matrix leads to good notions of critical and superstable configurations. They further generalized this model by introducing an invertible matrix $L$, calling $(L,M)$ a \emph{chip-firing pair}, and extending the definition of critical and superstable configurations to that model \cite{GuzKlivans}.

%%%TODO : add GK (L,M)-pair paper as reference

Our goal is to extend the duality between critical and superstable configurations to $(L,M)$-chip firing pairs. This will answer Question~5.2 of \cite{cho2024} in a much more general sense, since signed graphs are special cases of the chip-firing pair model. In Section~$2$ we summarize the Guzm\'an--Klivans theory of chip-firing pairs and also review the previously known duality between superstable and critical configurations of $M$-matrices. In Section~$3 $ we provide our main result on extending the duality map to chip-firing pairs. In Section~$4$ we study some properties of the map discussed in the main result. %then study some properties of this map.


%An important subtlety here is that one must consider \emph{multiset} firings and the notion of `$z$-superstability', we review these concepts below. 

\section{Prerequisites}
In this section we review the definition of chip-firing pairs. After that we review the duality map between superstable configurations and critical configurations for $M$-matrices. Then we will go over the tools developed in \cite{cho2024} that we will use to deal with the $z$-superstable and critical configurations of chip-firing pairs.

\subsection{Chip firing pairs}
In \cite{GuzKliMmatrices}, Guzm\'an and Klivans generalized the chip-firing on graphs to \newword{$M$-matrices}. $M$-matrices are used in various fields such as economics or scientific computing \cite{BurmanErn05,CiarletRaviart73,Leontief41,Plemmons77}. Guzm\'an and Klivans further generalized this by introducing an invertible integer matrix $L$ in \newword{chip-firing pairs} $(L,M)$ introduced in \cite{GuzKlivans}. 

\begin{definition}
\label{def:Mmatrix}
Suppose $M$ is an $n \times n$ matrix such that $(M)_{ii} > 0$ for all $i$ and $(M)_{ij} \leq 0$ for all $i \neq j$. Then $M$ is called an (invertible) \emph{$M$-matrix} if any of the following equivalent conditions hold:
\begin{itemize}
    \item $M$ is avalanche finite;
    \item The real part of the eigenvalues of $M$ are positive;
    \item The entries of $M^{-1}$ are non-negative;
    \item There exists a vector $\vec{x} \in {\mathbb R}^{n}$ with ${\vec x} \geq \vec{0}$ such that $M \vec{x}$ has all positive entries.
\end{itemize}
\end{definition}

The pair $(L,M)$, an $M$-matrix $M$ together with an invertible integer matrix $L$, is called a \emph{chip-firing pair}. The relevant (chip) \emph{configurations} $\vec{c} \in \mathbb{Z}^{n}$ are simply integer vectors with $n$ entries, and chip-firing is dictated by the matrix $L$. In particular, $(M,M)$ recovers the chip-firing on $M$-matrices and $(L_G,L_G)$ when $L_G$ is the (reduced) Laplacian of a graph recovers the classical chip-firing model on graphs.

\begin{remark}
In this paper, we only focus on integral $M$-matrices (since the duality map for $M$-matrices given in \cite{GuzKliMmatrices} that we extend upon, is only given for integral matrices).
\end{remark}



\begin{definition}\label{defn:valid}
Suppose $(L,M)$ is a chip-firing pair.  A configuration $\vec{c}$ is \emph{valid} if $\vec{c} \in S^+$, where
\[S^+ = \{LM^{-1} \vec{x} : LM^{-1}\vec{x} \in \mathbb{Z}^{n}, \vec{x} \in \mathbb{R}^n_{\geq 0}\}. \]

Equivalently, a configuration $\vec{c}$ is valid if $ML^{-1}\vec{c} \in R^+$, where
\[R^+ = \{ \vec{x} \in \mathbb{R}^n_{\geq 0} : LM^{-1}\vec{x} \in \mathbb{Z}^{n} \}.\]
\end{definition}

In particular, for $(M,M)$, being valid is the same as being a nonnegative integer vector. 

\begin{definition}
Suppose $(L,M)$ is a chip-firing pair, and suppose that $\vec{c} \in S^+$ is a valid configuration. A site $i \in \{1,\dots, n\}$ is \emph{ready to fire} if
\[\vec{c} - L\vec{e}_i \in S^+,\]
so that the vector obtained by subtracting the $i$th row of $L$ from $\vec{c}$ is also valid.

Similarly, suppose $\vec{x} \in R^+$. Then a site $i \in \{1,\dots, n\}$ is \emph{ready to fire} if 
\[\vec{x} - M\vec{e}_i \in R^+.\]

A configuration $\vec{c}$ (in $S^+$ or $R^+$) is \emph{stable} if no site is ready to fire.
\end{definition}

If $i$ is ready to fire, we declare that $\vec{b} = \vec{c} - L \vec{e}_i \in S^+$ is derived from $\vec{c}$ through a \emph{legal firing}. Repeating this process, a vector $\vec{a} \in S^+$ is said to be derived from $\vec{c}$ through a \emph{sequence of legal firings}. For a configuration $\vec{c} \in S^+$ (or conversely, $\vec{d} \in R^+$), we define $\text{stab}_{S^+}(\vec{c})$ (and $\text{stab}_{R^+}(\vec{d})$) as the resulting configuration after executing a series of legal firings until no site remains eligible to fire. Adapting the proof presented in \cite[Theorem 2.2.2]{Klivans}, it can be established that both $\text{stab}_{S^+}(\vec{c})$ and $\text{stab}_{R^+}(\vec{d})$ are uniquely determined. When it is clear whether we are dealing with $S^{+}$ or $R^{+}$, we use $\text{stab}(\vec{x})$ to refer to the stabilization of the configuration $\vec{x}$. 








%In this context the notion of a \emph{valid} configuration takes on a new meaning.



The definition of critical and superstable configurations in this model are as follows.


\begin{definition}
        Given an $(L,M)$ pair, a configuration $\vec c \in S^+$ is reachable if there exists some configuration $\vec d \in S^+$ satisfying:

        \begin{itemize}
            \item $\vec d - L \vec e_i \in S^+$ for all $1 \leq i \leq n$
            \item $\vec c = \vec d - \sum_{j = 1}^k L \vec e_j$ and $\vec d - \sum_{j = 1}^\ell L \vec e_j \in S^+$ for all $\ell < k$.
        \end{itemize}
        Given an $(L,M)$ pair, a configuration $\vec c \in S^+$ is critical if $\vec c$ is both stable and reachable.
    \end{definition}

Critical configurations are those that are both stable and reachable by chip-firing from a sufficiently large configuration. They are useful because they index the equivalence classes of the critical group.

\begin{definition}[{\cite[Definition 4.3]{GuzKliMmatrices}}]
A vector $f \in \ZZ^n$ with $f \geq 0$ is \newword{$z$-superstable} if for every $z \in \ZZ^n$ with $z \geq 0$ and $z \neq 0$ there exists $1 \leq i \leq n$ such that $f_i - (Lz)_i < 0$.
\end{definition}

It turns out that in the equivalence class given by the matrix $L$ in $S^{+}$, we can always find a unique representative that is critical and a unique representative that is $z$-superstable.

\begin{theorem}[{\cite[Theorems 3.5, 4.3, 5.5]{GuzKlivans}}]\label{thm:class}
Suppose $(L,M)$ is a chip-firing pair.  Then there exists exactly one $z$-superstable configuration and one critical configuration in each equivalence class $[\vec{c}]_L$.
\end{theorem}

\begin{remark}
For chip-firing pairs, there is the notion of a $\chi$-superstable configuration as well as a $z$-superstable configuration. From \Cref{thm:class}, it is the $z$-superstable configurations that have the same size as the critical configurations. For the remainder of the paper, we will focus only on $z$-superstable configurations, and we will call them the superstable configurations of the chip-firing pair, omitting the letter $z$.
\end{remark}


In the next subsection, we go over the duality that is known to exist when $L=M$.

\subsection{Duality for \texorpdfstring{$M$}{M}-matrices}

If we take a chip-firing pair $(M,M)$, it recovers the chip-firing on $M$-matrices studied in \cite{GuzKliMmatrices}. Chip-firing on $M$-matrices generalizes many properties and results of the classical chip-firing on graphs, and one of them is the duality between superstable and critical configurations.

Given an $M$-matrix, it turns out that there is a critical configuration that is a 
{\parfillskip=0pt \par} \noindent
coordinate-wise greater or equal to every other critical configuration. We call this configuration $\cm$, given by taking all diagonal entries of $M$ minus one and forming a vector (in the classical case, this corresponds to having $\deg(v)-1$ chips for each vertex $v$). 

\begin{theorem}[\cite{GuzKliMmatrices}]
\label{thm:usualMduality}
Let $M$ be an $M$-matrix. Let $\cm$ denote the vector where each entry is coming from the corresponding diagonal entry $M_{ii}$ minus one. Then we have a bijection between superstable and critical configurations by the map $\vec{c} \rightarrow \cm - \vec{c}$.
\end{theorem}


    % go over crit/ superstable definitions again
    % go over critical group again
 
   \begin{example}
    \label{ex:unsignedkyle}
    %TODO : Write down L_G
    Consider the following graph that has the reduced Laplacian to be \\
    \begin{minipage}{0.6\linewidth}
            $$L_G = \begin{pmatrix}
        3 & -1 & -1 \\
        -1 & 2 & -1 \\
        -1 & -1 & 3
    \end{pmatrix}:$$
        \end{minipage}
    \begin{minipage}{0.3\linewidth}
            \begin{center}
            \begin{tikzpicture}[scale=2]
            \input{tikz/kyle-unsigned}
            \end{tikzpicture}
            \end{center}
        \end{minipage}
        
    The superstable configurations and critical configurations are given in the following table:
    %In the table below, we can see that $\flip$ gives us a bijection between the superstables and criticals.
   \begin{center}\label{table:unsigned_toy_ss}
    \begin{tabular}{ c | c } 
        Superstables & Criticals \\
        \hline
        $(0, 0, 0)$ & $(2, 1, 2)$\\ 
        $(0, 0, 1)$ & $(2, 1, 1)$\\  
        $(0, 0, 2)$ & $(2, 1, 0)$\\
        $(0, 1, 0)$ & $(2, 0, 2)$\\
    \end{tabular}
    \begin{tabular}{ c | c }
        Superstables & Criticals \\
        \hline
        $(0, 1, 1)$ & $(2, 0, 1)$\\
        $(1, 0, 0)$ & $(1, 1, 2)$\\
        $(1, 1, 0)$ & $(1, 0, 2)$\\
        $(2, 0, 0)$ & $(0, 1, 2)$ 
    \end{tabular}
    \end{center}
    \end{example}

Notice that in the above example, we have a bijection between superstable configurations and critical configurations via the map $\vec{c} \rightarrow (2,1,2) - \vec{c}$. 


Recall that our goal is to extend this to $(L,M)$ chip-firing pairs. Next subsection will show that the map $\vec{c} \rightarrow \cmax - \vec{c}$ does not work for chip-firing pairs.


    \subsection{The usual duality map does not work for chip-firing pairs}

Recall that the usual duality map between the superstable and 
critical configurations for graphs (and also $M$-matrices) is given by the map $\vec{c} \rightarrow \cm - \vec{c}$ for some fixed $\cm$. As can be seen in the example below, this does not work for $(L,M)$-pairs in general. The examples from this point throughout will be using $(L,M)$-pairs coming from a signed graph. The systematic study of signed graphs and their Laplacian was initiated by Zaslavsky in \cite{Zaslavsky} and also studied in \cite{MR666857,MR4641710,FundBapat}.

    \begin{example} \label{ex:signed_kyle} \label{table:signed_toy_ss}
    We take $L$ to be the (reduced) Laplacian of the following signed graph and $M$ to be the (reduced) Laplacian of the underlying unsigned graph. The Laplacian of the signed graph is simply obtained from the Laplacian of the underlying graph, by changing the signs of entries corresponding to negative edges. 

    %%PUT the matrix L here
    \begin{figure}[ht]
        \centering
        \begin{minipage}{0.3\linewidth}
            \begin{center}
            \begin{tikzpicture}[scale=2]
            \input{tikz/kyle-signed.tex}
            \end{tikzpicture}
            \end{center}
        \end{minipage}
        \begin{minipage}{0.6\linewidth}
            $$M = \begin{pmatrix}
        3 & -1 & -1 \\
        -1 & 2 & -1 \\
        -1 & -1 & 3
    \end{pmatrix} \hspace{5mm}
    L = \begin{pmatrix}
        3 & 1 & -1 \\
        1 & 2 & -1 \\
        -1 & -1 & 3
    \end{pmatrix}$$
        \end{minipage}
        \label{fig:Signed-Kyle-S+}
    \end{figure}
    

    

    
    

    %%Instead of preimages put configs of $S^+$
    
    \begin{center}
    Configurations in $S^+$
    
    \renewcommand{\arraystretch}{1.2}
    \begin{tabular}{ c | c } \label{table:toy_ss}
        Superstables & Criticals \\
        \hline
        $(0, 0, 0) $ & $(6, 4, 2) $ \\
        $(1, 1, 0) $ & $(7, 5, 2) $ \\	
        $(4, 3, 2) $ & $(8, 6, 0) $ \\
        $(5, 4, 2) $ & $(9, 7, 0) $ \\
    \end{tabular}
    \begin{tabular}{c | c}
    Superstables & Criticals \\\hline
        $(2, 2, 0) $ & $(8, 6, 2) $ \\	
        $(3, 3, 0) $ & $(9, 7, 2) $ \\	
        $(3, 2, 0) $ & $(6, 4, 1) $ \\
        $(4, 3, 0) $ & $(7, 5, 1) $ \\
    \end{tabular}
    \begin{tabular}{c | c}
    Superstables & Criticals \\\hline
        $(5, 4, 0) $ & $(8, 6, 1) $ \\
        $(6, 5, 0) $ & $(9, 7, 1) $ \\
        $(6, 4, 0) $ & $(6, 5, 2) $ \\
        $(7, 5, 0) $ & $(7, 6, 2)$
    \end{tabular}
    \end{center}


    \end{example}

Notice that in the table of superstable and critical configurations of the chip-firing pair, the coordinate-wise maximal critical configuration is $(9,7,2)$. However if we take the superstable configuration $(5,4,2)$, the vector we get by applying the traditional duality map $(9,7,2) - (5,4,2) = (4,3,0)$ is not a critical configuration.

Even worse, there are many cases where $c_{max}$, the critical configuration that has coordinate-wise maximal entries does not even exist.


\begin{example}
\label{ex:nocmax}
Consider the $(L,M)$-pair coming from the signed graph below. The underlying graph is $C_6$, the cycle on six vertices.

\begin{figure}[ht]
    \centering
    \renewcommand{\arraystretch}{1.2}
    \begin{minipage}{.45\linewidth}
        \centering
        \begin{tikzpicture}[scale=.85]
            \input{tikz/c6-signed}
        \end{tikzpicture}    
    \end{minipage}
    \begin{minipage}{.45\linewidth}
        \centering
        \begin{tabular}{ c } %\label{table:toy_ss}
        Critical configurations \\
        \hline
        (9, 15, 17, 15, 9) \\
        (12, 20, 23, 21, 13) \\
        (13, 21, 23, 20, 12) \\
        (7, 11, 12, 11, 7) \\
        (10, 16, 18, 17, 11) \\
        (11, 17, 18, 16, 10) \\
        \end{tabular}
    \end{minipage}
    \label{fig:Cycle-5}
\end{figure}

As can be checked from the table of critical configurations above, there is no critical configuration that is the maximal in all coordinates. 
\end{example}





\subsection{Finding the superstable/critical configurations of chip firing pairs.}

In this subsection, we discuss an alternative way to find the superstable and critical configurations of $(L,M)$ chip-firing pairs, developed in \cite{cho2024}. %Moreover, from this point, whenever we say superstable configuration for $(L,M)$-pairs, we mean $z$-superstable configurations.

Let $\sstab(M)$ denote the set of superstable configurations of a $M$-matrix $M$ and let $\crit(M)$ denote the set of critical configurations. However, beware that we are not going to be using $\sstab(L,M)$ to denote the set of superstable configurations of $(L,M)$ and the same for $\crit(L,M)$. It turns out that for configurations in $S^{+}$, it is important to look at their preimages in $R^{+}$. Given any vector $\vec{f}$, we use $\floor{\vec{f}}$ to denote the vector obtained from $f$ by taking the floor at every coordinate.

    \begin{theorem}[{\cite[Theorem 3.2]{cho2024}}]
    \label{thm:22floor}
        Given an $(L, M)$ pair, a configuration $\vec c \in S^+$ is superstable/critical if and only if $\floor{ML^{-1} \vec c}$ is a  superstable/critical configuration of $M$.
    \end{theorem}

For example, we look at our running example coming from a signed graph.
%take a look at the the chip-firing pair coming from a signed graph in \Cref{table:signed_toy_ss}. 

    \begin{example}
    \label{ex:usualbad}
    Consider the signed graph studied in  \Cref{table:signed_toy_ss}. The table lists all superstable and critical configurations in $S^{+}$, their preimage in $R^{+}$, and the floor of the preimage. We can notice that the floor of the preimages are the superstable and critical configurations of the underlying graph we saw in \Cref{ex:unsignedkyle} (however, not all superstable/critical configurations of the underlying graph are used).
    
    %the preimages of all the superstable configurations and critical configurations, and their floors. %It is easy to check that a vector $\vec{c}$ such that $\vec{v} %\rightarrow 
    %\vec{c} - \vec{v}$ gives a bijection between the superstables and criticals does not exists here: since we have $(0,0,0)$ as a superstable and $(8/3,11/6,0)$ is a maximal configuration (coordinate-wise) we must have $\vec{c} = (8/3,11/6,0)$, but then $(2/3,1/3,2)$ does not even map to a vector with nonnegative entries.

    %We can see that each superstable in $R^+$ floors to an unsigned superstable for this example. The same is true for the criticals. 
    
    \begin{center}
        \begin{tikzpicture}[scale=2]
            \input{tikz/kyle-signed}
        \end{tikzpicture}
    \end{center}


    %%% Superstable in S^{+}, preimage in R^{+}, floor, crit in S^{+}, preimage in R^{+}, floor.
    \begin{center}\label{table:toy_stats}
    \renewcommand{\arraystretch}{1.2}
    \begin{tabular}{ c | c | c | c | c | c}
$\LM(\sstab)$ & $\sstab$                         & $\floor{\sstab}$ & $\LM(\crit)$& $\crit$                             & $\floor{\crit}$\\\hline
$ (0, 0, 0) $ & $(0, 0, 0) $                     & $ (0, 0, 0) $    & $(6,4,2)   $& $ (2, 0, 2) $                       & $ (2, 0, 2) $ \\
$ (1, 1, 0) $ & $(0, \nicefrac{1}{2}, 0) $           & $ (0, 0, 0) $    & $(7,5,2)   $& $ (2, \nicefrac{1}{2}, 2) $             & $ (2, 0, 2) $ \\
$ (4, 3, 2) $ & $(\nicefrac{2}{3}, \nicefrac{1}{3}, 2) $ & $ (0, 0, 2) $    & $(8,6,0)   $& $ (\nicefrac{8}{3}, \nicefrac{4}{3}, 0) $   & $ (2, 1, 0) $ \\
$ (5, 4, 2) $ & $(\nicefrac{2}{3}, \nicefrac{5}{6}, 2) $ & $ (0, 0, 2) $    & $(9,7,0)   $& $ (\nicefrac{8}{3}, \nicefrac{11}{6}, 0)  $ & $ (2, 1, 0) $ \\
$ (2, 2, 0) $ & $(0, 1, 0) $                     & $ (0, 1, 0) $    & $(8,6,2)   $& $ (2, 1, 2) $                       & $ (2, 1, 2) $ \\
$ (3, 3, 0) $ & $(0, \nicefrac{3}{2}, 0) $           & $ (0, 1, 0) $    & $(9,7,2)   $& $ (2, \nicefrac{3}{2}, 2)  $            & $ (2, 1, 2) $ \\
$ (3, 2, 0) $ & $(\nicefrac{4}{3}, \nicefrac{1}{6}, 0) $ & $ (1, 0, 0) $    & $(6,4,1)   $& $ (\nicefrac{7}{3}, \nicefrac{1}{6}, 1)  $  & $ (2, 0, 1) $ \\
$ (4, 3, 0) $ & $(\nicefrac{4}{3}, \nicefrac{2}{3}, 0) $ & $ (1, 0, 0) $    & $(7,5,1)   $& $ (\nicefrac{7}{3}, \nicefrac{2}{3}, 1) $   & $ (2, 0, 1) $ \\
$ (5, 4, 0) $ & $(\nicefrac{4}{3}, \nicefrac{7}{6}, 0) $ & $ (1, 1, 0) $    & $(8,6,1)   $& $ (\nicefrac{7}{3}, \nicefrac{7}{6}, 1) $   & $ (2, 1, 1) $ \\
$ (6, 5, 0) $ & $(\nicefrac{4}{3}, \nicefrac{5}{3}, 0) $ & $ (1, 1, 0) $    & $(9,7,1)   $& $ (\nicefrac{7}{3}, \nicefrac{5}{3}, 1) $   & $ (2, 1, 1) $ \\
$ (6, 4, 0) $ & $(\nicefrac{8}{3}, \nicefrac{1}{3}, 0) $ & $ (2, 0, 0) $    & $(6,5,2)   $& $ (\nicefrac{2}{3}, \nicefrac{4}{3}, 2) $   & $ (0, 1, 2) $ \\
$ (7, 5, 0) $ & $(\nicefrac{8}{3}, \nicefrac{5}{6}, 0) $ & $ (2, 0, 0) $    & $(7,6,2)   $& $ (\nicefrac{2}{3}, \nicefrac{11}{6}, 2) $  & $ (0, 1, 2) $
    \end{tabular}
    \end{center}
    \end{example}



Thanks to \Cref{thm:22floor}, it is much more convenient to deal with the preimages of configurations, especially when trying to check if it is superstable or critical. Given a superstable configuration in $S^{+}$, we will denote its preimage in $R^{+}$ as \newword{superstable preimage} and for a critical configuration in $S^{+}$, we are going to denote its preimage in $R^{+}$ as \newword{critical preimage}. We are also going to use $\sstab(L,M)$ to denote the set of superstable preimages of a $(L,M)$ chip-firing pair and use $\crit(L,M)$ to denote the set of critical preimages.

    %\textbf{sstab}:
    %Given an $(L, M)$ pair, the set $\sstab(L)$ is defined as the set of all signed superstable configurations, and the set $\sstab(M)$ denotes the set of all unsigned superstable configurations. If $\vec c$ is an unsigned/signed configuration, then $\sstab(\vec c)$ denotes the unique unsigned/signed superstable configuration that is firing-equivalent to $\vec c$.

    %\textbf{crit}: Given an $(L, M)$ pair, the set $\crit(L)$ is defined as the set of all signed critical configurations, and the set $\crit(M)$ denotes the set of all unsigned critical configurations. If $\vec c$ is an unsigned/signed configuration, then $\sstab(\vec c)$ denotes the unique unsigned/signed critical configuration that is firing-equivalent to $\vec c$.



%Given an $(L,M)$ chip-firing pair, we let $\sstab(L,M)$ denote the set of ($z$-)superstable configurations and $\crit(L,M)$ denote the set of critical configurations. 
      
      %Given a graph $G$ with a sink $q$, with  we define its reduced Laplacian matrix $L_G$ as
    %\[(L_G)_{ij} = \begin{cases}
     %   \deg(v_i) & i = j  \\
      %  -1 & (ij) \in E(G) \\
        %0 & \text{ otherwise,}
    %\end{cases}\]
    %where the columns and rows are indexed with $V(G) \setminus q$. 

    
    %Configurations on $G$ are represented by vectors $\cc \in \ZZ^{|G \backslash \{q\}|}$, %todo not sure if this is standard
    %where the $i$th coordinate of $\cc$ corresponds to the number
%    of chips on the $i$th vertex of $G$.

    %It turns out that $L_G$ is an $M$-matrix, and since it is also invertible, we can study the chip-firing pairs $(L_G,L_G)$. Then the definitions we went over in the previous subsection align with the classical definitions for chip-firing on graphs. In particular, the $z$-superstable configurations of $(L_G,L_G)$ are exactly the superstable configurations of $G$.


    %We define the reduced Laplacian $L_{G_\phi}$ of a signed graph $G_\phi$ in the same way as the reduced Laplacian of an unsigned graph, except that entries in the matrix corresponding to negative edges take the value $1$ (as opposed to $-1$ for positive edges). Defining the reduced Laplacian this way allows us to inherit the same chip-firing rule from unsigned graphs, where chip-firing vertex $v$ from configuration $\vec{c}$ results in the configuration $\vec{c} - L_{G_\phi} \vec{e}_v$. The reduced Laplacian also defines a critical group $\mathcal{K}(L_{G_\phi})$. We also refer to this group as simply $\mathcal{K}(L)$ when it is clear from context which signed graph the group corresponds to, and we refer to the reduced Laplacian as simply $L$.

    %We take these definitions from \cite{cho2024}, and hereafter we use "$z$-superstable" and "superstable" interchangeably, as Theorem 3.6 in \cite{cho2024} shows that given an arbitrary $(L,M)$ pair, the set of $z$-superstable and $\chi$-superstable configurations are equivalent.

    % TODO: where is this result from?



    %When considering signed graphs, there is no bijection between signed superstable and critical configurations that is as easy to describe as the duality on unsigned graphs. We investigate this problem in the next section.


    %TODO the outline for the pdf at this part looks super weird --   