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\pagestyle{fancy} \lhead{{\sf Homework 8 $|$ Due November 17 (Monday)
}} \rhead{\thepage} \cfoot{{\sf Math 4110 $|$ Combinatorics
    $|$ Fall 2025 $|$ M.~Macauley}}


\begin{document}

%$\;$

\noindent\textbf{Topics}: Chip firing


\begin{enumerate}


\item Let $G=K_4\setminus\{1,3\}$. The eight critical chip configurations are shown below, along with the result of Dhar's burning test, and the corresponding spanning tree.
\[
\begin{tikzpicture}[
  scale=.8,
  >=Stealth,
  line width=.8pt,
  r/.style={circle, draw, fill=red!30, inner sep=0pt, minimum size=3.75mm,font=\scriptsize},
  w/.style={circle, draw, inner sep=0pt, minimum size=3.75mm,font=\scriptsize},
  e/.style={gray},
  burned/.style={red,very thick},
  every label/.style={font=\tiny}
]
%%--------------------------------------------------------
\begin{scope}[shift={(0,3)}]
\node[r] (b) at (90:1) {$1$}; 
\node[r] (a) at (180:1) {$0$}; \node[r] (c) at (0:1) {$1$}; 
\node[r] (d) at (270:1) {\tiny$-\!2$}; 
\draw[burned] (a) to (b); \draw[burned] (b) to (c); 
\draw[burned] (c) to (d); \draw[e] (d) to (a); 
\draw[e] (b) to (d); 
\end{scope}
%%
\begin{scope}[shift={(3.5,3)}]
\node[r] (b) at (90:1) {$2$}; 
\node[r] (a) at (180:1) {$0$}; \node[r] (c) at (0:1) {$0$}; 
\node[r] (d) at (270:1) {\tiny$-\!2$}; 
\draw[burned] (a) to (b); \draw[burned] (b) to (c); 
\draw[e] (c) to (d); \draw[e] (d) to (a); 
\draw[burned] (b) to (d); 
\end{scope}
\begin{scope}[shift={(7,3)}]
\node[r] (b) at (90:1) {$2$}; 
\node[r] (a) at (180:1) {$0$}; \node[r] (c) at (0:1) {$1$}; 
\node[r] (d) at (270:1) {\tiny$-\!2$}; 
\draw[burned] (a) to (b); \draw[e] (b) to (c); 
\draw[burned] (c) to (d); \draw[e] (d) to (a); 
\draw[burned] (b) to (d); 
\end{scope}
\begin{scope}[shift={(10.5,3)}]
\node[r] (b) at (90:1) {$0$}; 
\node[r] (a) at (180:1) {$1$}; \node[r] (c) at (0:1) {$1$}; 
\node[r] (d) at (270:1) {\tiny$-\!2$}; 
\draw[e] (a) to (b); \draw[burned] (b) to (c); 
\draw[burned] (c) to (d); \draw[burned] (d) to (a); 
\draw[e] (b) to (d); 
\end{scope}
%%-------------------------------------
\begin{scope}[shift={(0,0)}]
\node[r] (b) at (90:1) {$1$}; 
\node[r] (a) at (180:1) {$1$}; \node[r] (c) at (0:1) {$0$}; 
\node[r] (d) at (270:1) {\tiny$-\!2$}; 
\draw[burned] (a) to (b); \draw[burned] (b) to (c); 
\draw[e] (c) to (d); \draw[burned] (d) to (a); 
\draw[e] (b) to (d); 
\end{scope}
%%
\begin{scope}[shift={(3.5,0)}]
\node[r] (b) at (90:1) {$1$}; 
\node[r] (a) at (180:1) {$1$}; \node[r] (c) at (0:1) {$1$}; 
\node[r] (d) at (270:1) {\tiny$-\!3$}; 
\draw[burned] (a) to (b); \draw[e] (b) to (c); 
\draw[burned] (c) to (d); \draw[burned] (d) to (a); 
\draw[e] (b) to (d); 
\end{scope}
%%
\begin{scope}[shift={(7,0)}]
\node[r] (b) at (90:1) {$2$}; 
\node[r] (a) at (180:1) {$1$}; \node[r] (c) at (0:1) {$0$}; 
\node[r] (d) at (270:1) {\tiny$-\!3$}; 
\draw[burned] (a) to (b); \draw[burned] (b) to (c); 
\draw[burned] (c) to (d); \draw[e] (d) to (a); 
\draw[e] (b) to (d); 
\end{scope}
%%
\begin{scope}[shift={(10.5,0)}]
\node[r] (b) at (90:1) {$2$}; 
\node[r] (a) at (180:1) {$1$}; \node[r] (c) at (0:1) {$1$}; 
\node[r] (d) at (270:1) {\tiny$-\!4$}; 
\draw[e] (a) to (b); \draw[e] (b) to (c); 
\draw[burned] (c) to (d); \draw[burned] (d) to (a); 
\draw[burned] (b) to (d); 
\end{scope}
\end{tikzpicture}
\]
Recall that the critical configurations form a group called the \emph{critical group}, denoted $\mathcal{K}(G)$, where the operation is defined as 
\[
[\c_1]+[\c_2]=[\c_1+\c_2].
\]
\begin{enumerate}
    \item The reduced Laplacian is $L_0(G)=\begin{bmatrix}2 & -1 & 0 \\ -1 & 3 & -1 \\
0 & -1 & 2 \end{bmatrix}$, and its determinant is $d_3=8$. Compute the greatest common divisor of all $2\times 2$ minors; call this $d_2$. Then compute $d_1$, the GCD of all $1\times 1$ minors. The sequence $d_1\mid d_2\mid d_3$ are the \emph{elementary divisors} of the so-called \emph{Smith normal form}, and they determine the critical group.

\item What is the identity element of $\mathcal{K}(G)$? That is, what chip configuration $\c_0$ satisfies $[\c_0]+[\c]=[\c]$ for every $[\c]\in\mathcal{K}(G)$?

\item Which (non-identity) chip configurations are their own inverses? That is, which satisfy $[\c]+[\c]=[\c]$? Use this to determine whether $\mathcal{K}(G)$ is isomorphic to $\Z_8$, $\Z_4\times\Z_2$, or $\Z_2\times\Z_2\times\Z_2$. 

\item Construct an $8\times 8$ \emph{Cayley table} for $\mathcal{K}(G)$. 

\item Construct a \emph{Cayley graph} of $\mathcal{K}(G)$, and put the chip configurations in the corresponding nodes. It will have the form of one of the following

\[
  \definecolor{lightgray}{rgb}{.85, .85, .85}
  \begin{tikzpicture}[scale=1,
  >=Stealth,
  r/.style={red,very thick,->},
  rr/.style={red,very thick},
  bb/.style={blue,very thick},
  gg/.style={black!30!green,very thick},
  R2/.style={red,very thick,->,bend right=15},
]
    \begin{scope}[shift={(0,2.25)}]
    \tikzstyle{v} = [circle, draw, fill=lightgray,inner sep=0pt, 
  minimum size=5.5mm]
      %%
      \node (1) at (0:2) [v] {$0$};
      \node (r) at (45:2) [v] {$1$};
      \node (r2) at (90:2) [v] {$2$};
      \node (r3) at (135:2) [v] {$3$};
      \node (r4) at (180:2) [v] {$4$};
      \node (r5) at (225:2) [v] {$5$};
      \node (r6) at (270:2) [v] {$6$};
      \node (r7) at (315:2) [v] {$7$};
      %%
      \draw [R2] (1) to (r);
      \draw [R2] (r) to (r2);
      \draw [R2] (r2) to (r3);
      \draw [R2] (r3) to (r4);
      \draw [R2] (r4) to (r5);
      \draw [R2] (r5) to (r6);
      \draw [R2] (r6) to (r7);
      \draw [R2] (r7) to (1);
    \end{scope}
    %%
    \begin{scope}[shift={(4.75,0)},scale=1.5]
    \tikzstyle{v} = [circle, draw, fill=lightgray,inner sep=0pt, 
  minimum size=6mm]
      \tikzstyle{every node}=[font=\scriptsize]
      \node (00) at (0,3) [v] {$(0,\!0)$};
      \node (10) at (0,2) [v] {$(1,\!0)$};
      \node (20) at (0,1) [v] {$(2,\!0)$};
      \node (30) at (0,0) [v] {$(3,\!0)$};
      \node (01) at (1,3) [v] {$(0,\!1)$};
      \node (11) at (1,2) [v] {$(1,\!1)$};
      \node (21) at (1,1) [v] {$(2,\!1)$};
      \node (31) at (1,0) [v] {$(3,\!1)$};
      \draw [r] (00) to (10);
      \draw [r] (10) to (20);
      \draw [r] (20) to (30);
      \draw [r] (30) to [bend left=25] (00);
      \draw [r] (01) to (11);
      \draw [r] (11) to (21);
      \draw [r] (21) to (31);
      \draw [r] (31) to [bend right=25] (01);
      \draw [bb] (00) to (01);
      \draw [bb] (10) to (11);
      \draw [bb] (20) to (21);
      \draw [bb] (30) to (31);
\end{scope}
%%
    \begin{scope}[shift={(10,1.75)},scale=.75]
    \tikzstyle{every node}=[font=\footnotesize]
      \tikzstyle{v} = [circle, draw, fill=lightgray,inner sep=0pt, 
  minimum size=7.5mm]
      \node (000) at (0,0) [v] {$000$};
      \node (100) at (-1.5,-1.5) [v] {$100$};
      \node (010) at (3,0) [v] {$010$};
      \node (110) at (1.5,-1.5) [v] {$110$};
      \node (001) at (0,3) [v] {$001$};
      \node (101) at (-1.5,1.5) [v] {$101$};
      \node (011) at (3,3) [v] {$011$};
      \node (111) at (1.5,1.5) [v] {$111$};
      \draw [gg] (000) to (001);
      \draw [rr] (000) to (010);
      \draw [bb] (000) to (100);
      \draw [rr] (100) to (110); \draw [rr] (101) to (111);
      \draw [rr] (001) to (011);
      \draw [gg] (100) to (101); \draw [gg] (110) to (111);
      \draw [gg] (010) to (011);
      \draw [bb] (101) to (001); \draw [bb] (111) to (011);
      \draw [bb] (110) to (010);
    \end{scope}
  \end{tikzpicture}
\]
\end{enumerate}

%%------------------------------------------------------------------------------

\item Consider the ``house graph'' $G$ shown below:
\[
\begin{tikzpicture}[
  scale=1.25,
  >=Stealth,
  line width=.8pt,
  v/.style={circle, draw, fill=black, inner sep=0pt, minimum size=1mm,font=\footnotesize},
  w/.style={circle, draw, inner sep=0pt, minimum size=3.5mm,font=\scriptsize},
  every label/.style={font=\footnotesize}
]
%%--------------------------------------------------------
\begin{scope}[shift={(0,0)}]
\node[v,label=above:$5$] (v5) at (.5,1.707) {}; 
\node[v,label=left:$1$] (v1) at (0,1) {}; \node[v,label=right:$2$] (v2) at (1,1) {}; 
\node[v,label=left:$4$] (v4) at (0,0) {}; \node[v,label=right:$3$] (v3) at (1,0) {}; 
\draw (v1) to (v2); \draw (v2) to (v3); \draw (v3) to (v4); 
\draw (v4) to (v1); \draw (v1) to (v5); \draw (v2) to (v5); 
\end{scope}
\end{tikzpicture}
\]
\begin{enumerate}
    \item Find the Laplacian $L(G)$ and compute $\det(L_0(G))$.
    \item Construct all critical chip configurations. Which is the identity element of $\mathcal{K}(G)$?
    \item Run Dhar's burning test on the critical configurations and associate each with a spanning tree.
    \item Construct all superstable chip configurations.
    \item Construct a Cayley graph with the nodes labeled by the corresponding chip configurations. 
\end{enumerate}



\end{enumerate}

\end{document}
