\section{Examples}
\label{sec:examples}
Here we apply our main theorem to some special quivers,
and consider the $f$-vector, i.e.,
the sequence of the number $f_d$ of the faces of dimension $d$.

\subsection{Quivers related to forests}
First we consider the case of an asymmetric quiver $Q$ with no
undirected closed walk, i.e.,
the case where the underlying graph of $Q$ is a forest.
In this case, we have
\begin{align*}
 \numof{\ver(\DE(Q))} & = \numof{Q_1} = c(Q). 
\end{align*}
On the other hand, $Q$ has a rank function and we have
$\dim(\DE(Q))=c(Q)-1$ by \cref{thm:dim}. 
Hence  
  $\DE(Q)$ is a simplex of dimension $\numof{Q_1}-1=c(Q)-1$.
We also have $f_d=\binom{\numof{Q_1}}{d+1}$, and
 $\DE(R)$ is a face of $\DE(Q)$ for any lluf subquiver $R$ of $Q$.


%% \begin{example}
%%  Let $Q$ be an asymmetric quiver whose underlying graph is the Dynkin
%%  graph $A_{n+1}$.
%%  Denote $Q_0=\Set{0,1,\ldots,n}$ and
%%  $Q_1=\Set{e_1,\ldots,e_n}$ so that $e_{i}=(i-1,i)$ or $(i,i-1)$ for
%%  $i=1,\ldots,n$. 
%%  Then the directed edge polytope $\DE(Q)$ is
%%  an $(n-1)$-simplex.
%%  Hence $\DE(R)$ is a face for any luff proper subquiver $R$
%%  of $Q$.
%% \end{example}

Next we consider a forest $G=(G_0,G_1)$ and $\SE(G)$.
Since a quiver whose underlying graph is $G$ has a rank function,
it is a facet of $\SE(G)$.
Hence
$f_d=\binom{\numof{G_1}}{d+1}2^{d+1}$.


\subsection{Quivers related to cycles}

Here we consider a quiver related to the cycle with $m$ edges and $m$ vertices.
In this case,
we regard  the vertex set $Q_0$
as $\ZZ/m\ZZ=\Set{\overline{1},\ldots,\overline{m}=\overline{0}}$.
Moreover, we define $O^+$ to be the set 
$\Set{(\overline{i-1},\overline{i})|\overline{i}\in Q_0}$
of `clock-wise' edges,
and
 $O^-$ to be the set 
$\Set{(\overline{i+1},\overline{i})|\overline{i}\in Q_0}$
of `anticlock-wise' edges.

First we consder the quiver whose underlying graph is
a cycle.

\begin{example}
  Let $Q$ be an asymmetric quiver whose underlying graph is
  the cycle with $2n+1$ edges.
  In this case, $Q$ has no rank function.
  Hence $\DE(Q)$ is a $2n$ dimensional polytope with $2n+1$ vertices.
  Thus $\DE(Q)$ is a simplex of dimension $2n$.
\end{example}

Next consider the case of even cycle.
\begin{example}
 \label{example:asymmetriccycle}
 We consider  an asymmetric quiver $Q$
 whose underlying graph is an even cycle.
 Let $m=2n$.
 We define 
  $Q^{+}_1=Q_1\cap O^{+}$,
  $Q^{-}_1=Q_1\cap O^{-}$.
 If $\numof{Q^{+}_1}=\numof{Q^{-}_1}=n$, 
 then $Q$ has a rank function.
 It follows from  \cref{thm:dim} that
 $ \dim(\DE(Q))= 2n-2$.
 Since $\numof{\ver(\DE(Q))}=\numof{Q_{1}}=2n$, $\DE(Q)$ is not a
 simplex. 
 
 Let $R$ be a lluf subquiver of $Q$ whose directed edge polytope
 $\DE(R)$ is a facet of $\DE(Q)$.
 Since $R$ also has a rank function,
 we have 
 \[
 2n-3=\dim(\DE(R)) = \numof{Q_{0}}-\numof{\pi_{0}(R)}-1
 =2n-1-\numof{\pi_{0}(R)}, 
 \]
 which implies that $\numof{Q_{1}\setminus R_{1}}=2$.
 Let $Q_1\setminus R_1 = \Set{e',e''}$. 
 The acyclicity of $Q/R$ following from \cref{thm:face} allows us to
 assume that $e'\in Q^{+}_{1}$ and $e''\in Q^{-}_{1}$.
 This is a characterization of facets of $\DE(Q)$.
 
 For such a subquiver $R$, $\DE(R)$ is a simplex
 of dimension $2n-3$.
 Since faces of a simplex are in one-to-one correspondence to subsets of
 the vertex set, for a lluf subquiver $S$ of $Q$,
  $\DE(S)$ is a proper face of $\DE(Q)$ of dimension $d$
 if and only if $\numof{S_1\cap Q^{+}_1}<n$,
 $\numof{S_1\cap Q^{-}_1}<n$, and $\numof{S_1}=d+1$.
 Hence 
 \begin{align*}
  f_d=\binom{2n}{d+1}-2\binom{n}{d+1-n},
 \end{align*}
  where the binomial coefficient $\binom{m}{k}$ equals $0$ if $m<k$ or
 $k<0$. 
\end{example}



Next we consider a cycle $G$ and $\SE(G)$.
\begin{example}
 \label{example:evencycle}
 Let $C_{2n}$ be the cycle with $2n$ edges and  $Q=D(C_{2n})$.
 In this case, we have $Q_1=O^+\cup O^-$.
 The symmetric edge polytope $\SE(C_{2n})=\DE(Q)$ is a
 $(2n-1)$-dimensional polytope by \cref{thm:dim}.
 The faces of $\SE(C_{2n})$ can be determined as follows.

 By (\ref{SE3}) of \cref{cor:SE}, for a lluf subquiver $R$ of
 $Q$ with
 \[
 \dim(\DE(R))=\dim(\SE(C_{2n}))-1=2n-2,  
 \]
 $\DE(R)$ is a facet
 of $\SE(C_{2n})$ if and only if 
 $c(R)=c(Q)=2n-1$ and
 there exists a function $\rho:\ZZ/2n\ZZ\to\RR$ such that
 \[
  \rho(\overline{i-1})-\rho(\overline{i}) =
 \begin{cases}
  1 & ((\overline{i-1},\overline{i})\in R_{1}) \\
  -1 & ((\overline{i},\overline{i-1})\in R_{1}) \\
  0 & (\text{otherwise}),
 \end{cases}
 \]
 which implies that only one of $(\overline{i-1},\overline{i})$ or
 $(\overline{i},\overline{i-1})$ belongs to $R_{1}$ for each $i$.
 Let $R^{+}_{1}  = R_{1}\cap O^{+}$  and $R^{-}_{1} = R_{1}\cap O^{-}$.
 Then we have $\numof{R^{+}_{1}}=\numof{R^{-}_{1}}$.
 
 Since $\DE(R)$ is of dimension $2n-2$,
 \[
 \numof{R_{1}} = \numof{\ver(\DE(R))} \ge 2n-1.
 \]
 By the condition on $\rho$, we see that the underlying graph of $R$
 must be the whole $C_{2n}$.
 %% Thus we have $\numof{R^{+}_1}=\numof{R^{-}_1}=n$ and 
 %% $R^{+}_{1}\cap (-R^{-}_{1})=\emptyset$, where
 %% $-R^{-}_{1}=\Set{(\overline{i-1},\overline{i})|(\overline{i},\overline{i-1})\in
 %% R^{-}_{1}}$.
 %% In other words, facets of $\SE(C_{2n})$ are in bijective correspondence
 %% to subsets of cardinality $n$ in
 %% $Q_{1}=\Set{(\overline{i-1},\overline{i}),(\overline{i},\overline{i-1})
 %% | i=1,\ldots,2n}$.
 Hence
 facets of $\SE(C_{2n})$ are in bijective correspondence
 to subsets of cardinality $n$ in $O^+$.
 Hence we have
  \begin{align*}
    f_{2n-2} = \binom{2n}{n}.
  \end{align*}
 
 Note that facets of $\SE(C_{2n})$ are polytopes in
 \cref{example:asymmetriccycle}. 
 In particular, faces of codimension $2$ in $\SE(C_{2n})$ are simplices
 of dimension $(2n-3)$,
 which means that all faces of $\SE(C_{2n})$ except for facets are
 simplices. 
 In other words, for $d<2n-2$ and a lluf subquiver $R$ of $Q$,
 $\DE(R)$ is a face of dimension $d$ in $\SE(C_{2n})$
 if and only if $\numof{R_1}=d+1$,
 $\numof{R_1\cap O^{+}}<n$,
 $\numof{R_1\cap O^{-}}<n$, and $R^{+}_{1}\cap (-R^{-}_{1})=\emptyset$.
 Hence it follows from direct calculation that
  \begin{align*}
    f_d &=\binom{2n}{d+1}2^{d+1}.
  \end{align*}

 We remark that D'Ali, Delucchi, and Micha{\l}ek \cite{MR4458147} also
 performed the same computation based on the characterization of facets
 by Higashitani et al.~\cite{MR3949939}.
\end{example}


\begin{example}
 \label{example:oddcycle}
 Consider the case of an odd cycle $C_{2n+1}$.
 Let $Q=D(C_{2n+1})$.
 The symmetric edge polytope $\SE(C_{2n+1})=\DE(Q)$ is a
 $2n$-dimensional polytope by \cref{thm:dim}. 
 
 For a lluf subquiver $R$ of $Q$, suppose that $\dim(\DE(R))=2n-1$.
 By the same argument as in \cref{example:evencycle},
 $\DE(R)$ is a facet of $\SE(C_{2n+1})$ if and only if
 $c(R)=c(Q)$, 
 $\numof{R^{+}_1}=\numof{R^{-}_1}=n$,
 and $R^{+}_1\cap (-R^{-}_{1})=\emptyset$.
 %% Since $\numof{\ver(\DE(R))}=2n$, $\DE(R)$ is a simplex of dimension
 %% $2n-1$ and all faces of $\SE(C_{2n+1})$ are simplices.
 %% We also see that facets are in one-to-one correspondence to a pair
 %% $(E,e)$ of a subset $E$ of $Q^{+}_{1}$ of cardinality $n$ and an
 %% element $e\in Q^{-}_{1}\setminus(-E)$.
 Hence we have
  \begin{align*}
    f_{2n-1} = (n+1)\binom{2n+1}{n} = \frac{(2n+1)!}{n!n!}=
   (2n+1)\binom{2n}{n}. 
  \end{align*}
 
 Thus, for $d<2n-2$ and a lluf subquiver $R$ of $Q$,
 $\DE(R)$ is a face of $\SE(C_{2n+1})$ of dimension $d$
 if and only if $\numof{R_1}=d+1$,
 $\numof{R^{+}_1}<n$, $\numof{R^{-}_1}<n$, and 
 $R^{+}_1\cap (-R^{-}_{1})=\emptyset$.
 Hence it follows from direct calculation that
  \begin{align*}
    f_d &=\binom{2n+1}{d+1}2^{d+1}.
  \end{align*}
\end{example}

\subsection{Quivers of small size}
As a final remark, 
we give examples of $f$-vectors of
directed edge polytopes for connected quivers with three vertices.
%
The directed edge polytopes for the following quivers are 1-dimensional simplices:
\begin{gather*}
\begin{picture}(30,30)
\put (5, 25) {\makebox(0,0){\tiny$\bullet$}}
\put (25, 25) {\makebox(0,0){\tiny$\bullet$}}
\put (25, 5) {\makebox(0,0){\tiny$\bullet$}}
\put (5, 25) {\vector (1, 0) {20}}
\put (5, 25) {\vector (1, -1) {20}}
\end{picture}
\begin{picture}(30,30)
\put (5, 25) {\makebox(0,0){\tiny$\bullet$}}
\put (25, 25) {\makebox(0,0){\tiny$\bullet$}}
\put (25, 5) {\makebox(0,0){\tiny$\bullet$}}
\put (5, 25) {\vector (1, 0) {20}}
\put (25, 25) {\vector (0, -1) {20}}
\end{picture}
\begin{picture}(30,30)
\put (5, 25) {\makebox(0,0){\tiny$\bullet$}}
\put (25, 25) {\makebox(0,0){\tiny$\bullet$}}
\put (25, 5) {\makebox(0,0){\tiny$\bullet$}}
\put (5, 25) {\vector (1, 0) {20}}
\put (25, 5) {\vector (0, 1) {20}}
\end{picture}
\end{gather*}
%
The directed edge polytopes for the following quivers are 2-dimensional simplices:
\begin{gather*}
\begin{picture}(30,30)
\put (5, 25) {\makebox(0,0){\tiny$\bullet$}}
\put (25, 25) {\makebox(0,0){\tiny$\bullet$}}
\put (25, 5) {\makebox(0,0){\tiny$\bullet$}}
\put (5, 25) {\vector (1, 0) {20}}
\put (5, 25) {\vector (1, -1) {20}}
\put (25, 25) {\vector (0, -1) {20}}
\end{picture}
\begin{picture}(30,30)
\put (5, 25) {\makebox(0,0){\tiny$\bullet$}}
\put (25, 25) {\makebox(0,0){\tiny$\bullet$}}
\put (25, 5) {\makebox(0,0){\tiny$\bullet$}}
\put (5, 25) {\vector (1, 0) {20}}
\put (25, 25) {\vector (-1, 0) {20}}
\put (25, 25) {\vector (0, -1) {20}}
\end{picture}
\begin{picture}(30,30)
\put (5, 25) {\makebox(0,0){\tiny$\bullet$}}
\put (25, 25) {\makebox(0,0){\tiny$\bullet$}}
\put (25, 5) {\makebox(0,0){\tiny$\bullet$}}
\put (5, 25) {\vector (1, 0) {20}}
\put (25, 25) {\vector (0, -1) {20}}
\put (25, 5) {\vector (-1, 1) {20}}
\end{picture}
\begin{picture}(30,30)
\put (5, 25) {\makebox(0,0){\tiny$\bullet$}}
\put (25, 25) {\makebox(0,0){\tiny$\bullet$}}
\put (25, 5) {\makebox(0,0){\tiny$\bullet$}}
\put (5, 25) {\vector (1, 0) {20}}
\put (25, 25) {\vector (-1, 0) {20}}
\put (25, 5) {\vector (0, 1) {20}}
\end{picture}
\end{gather*}
%
Next we give examples of the case where directed edge polytopes are not simplices.
The $f$-vectors for the directed edge polytopes for the following quivers are $ (1, 4, 4, 1) $:
\begin{gather*}
\begin{picture}(30,30)
\put (5, 25) {\makebox(0,0){\tiny$\bullet$}}
\put (25, 25) {\makebox(0,0){\tiny$\bullet$}}
\put (25, 5) {\makebox(0,0){\tiny$\bullet$}}
\put (5, 25) {\vector (1, 0) {20}}
\put (5, 25) {\vector (1, -1) {20}}
\put (25, 25) {\vector (0, -1) {20}}
\put (25, 5) {\vector (-1, 1) {20}}
\end{picture}
\begin{picture}(30,30)
\put (5, 25) {\makebox(0,0){\tiny$\bullet$}}
\put (25, 25) {\makebox(0,0){\tiny$\bullet$}}
\put (25, 5) {\makebox(0,0){\tiny$\bullet$}}
\put (5, 25) {\vector (1, 0) {20}}
\put (5, 25) {\vector (1, -1) {20}}
\put (25, 25) {\vector (0, -1) {20}}
\put (25, 5) {\vector (0, 1) {20}}
\end{picture}
\begin{picture}(30,30)
\put (5, 25) {\makebox(0,0){\tiny$\bullet$}}
\put (25, 25) {\makebox(0,0){\tiny$\bullet$}}
\put (25, 5) {\makebox(0,0){\tiny$\bullet$}}
\put (5, 25) {\vector (1, 0) {20}}
\put (5, 25) {\vector (1, -1) {20}}
\put (25, 25) {\vector (-1, 0) {20}}
\put (25, 25) {\vector (0, -1) {20}}
\end{picture}
\begin{picture}(30,30)
\put (5, 25) {\makebox(0,0){\tiny$\bullet$}}
\put (25, 25) {\makebox(0,0){\tiny$\bullet$}}
\put (25, 5) {\makebox(0,0){\tiny$\bullet$}}
\put (5, 25) {\vector (1, 0) {20}}
\put (25, 25) {\vector (-1, 0) {20}}
\put (25, 25) {\vector (0, -1) {20}}
\put (25, 5) {\vector (0, 1) {20}}
\end{picture}
\end{gather*}
%
The $f$-vector for the directed edge polytope for the following quiver is $ (1, 5, 5, 1) $:
\begin{gather*}
\begin{picture}(30,30)
\put (5, 25) {\makebox(0,0){\tiny$\bullet$}}
\put (25, 25) {\makebox(0,0){\tiny$\bullet$}}
\put (25, 5) {\makebox(0,0){\tiny$\bullet$}}
\put (5, 25) {\vector (1, 0) {20}}
\put (5, 25) {\vector (1, -1) {20}}
\put (25, 25) {\vector (0, -1) {20}}
\put (25, 5) {\vector (-1, 1) {20}}
\put (25, 5) {\vector (0, 1) {20}}
\end{picture}
\end{gather*}
%
The $f$-vector for the directed edge polytope for the following quiver is $ (1, 6, 6, 1) $:
\begin{gather*}
\begin{picture}(30,30)
\put (5, 25) {\makebox(0,0){\tiny$\bullet$}}
\put (25, 25) {\makebox(0,0){\tiny$\bullet$}}
\put (25, 5) {\makebox(0,0){\tiny$\bullet$}}
\put (5, 25) {\vector (1, 0) {20}}
\put (5, 25) {\vector (1, -1) {20}}
\put (25, 25) {\vector (-1, 0) {20}}
\put (25, 25) {\vector (0, -1) {20}}
\put (25, 5) {\vector (-1, 1) {20}}
\put (25, 5) {\vector (0, 1) {20}}
\end{picture}
\end{gather*}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
In the case of the quivers with three vertices,
we have only three kinds of $f$-vectors for derected edge polytopes
which are not simplices.
In the case of the quivers with four vertices,
we, however, have more kinds of $f$-vectors
as follows:
\begin{align*}
   &(1, 5, 8, 5, 1),
  &&(1, 6, 10, 6, 1),
  &&(1, 7, 13, 8, 1),
  &&(1, 8, 16, 10, 1),
  &&(1, 9, 19, 12, 1),\\
   &(1, 9, 17, 10, 1),
  &&(1, 10, 20, 12, 1),
  &&(1, 6, 11, 7, 1),
  &&(1, 7, 14, 9, 1),
  &&(1, 8, 17, 11, 1),\\
   &(1, 8, 14, 8, 1),
  &&(1, 9, 15, 8, 1),
  &&(1, 7, 12, 7, 1),
  &&(1, 5, 9, 6, 1),
  &&(1, 7, 11, 6, 1),\\
   &(1, 9, 18, 11, 1),
  &&(1, 10, 21, 13, 1),
  &&(1, 6, 12, 8, 1),
  &&(1, 7, 15, 10, 1),
  &&(1, 6, 9, 5, 1),\\
   &(1, 10, 19, 11, 1),
  &&(1, 11, 22, 13, 1),
  &&(1, 12, 24, 14, 1),
  &&(1, 8, 13, 7, 1),
  &&(1, 4, 4, 1),\\
   &(1, 8, 18, 12, 1),
  &&(1, 8, 15, 9, 1),
  &&(1, 8, 12, 6, 1),
  &&(1, 9, 16, 9, 1).
\end{align*}

%% (1,4,4,1)
%% \begin{picture}(30,30)
%% \put (5, 25) {\makebox(0,0){\tiny$\bullet$}}
%% \put (25, 25) {\makebox(0,0){\tiny$\bullet$}}
%% \put (25, 5) {\makebox(0,0){\tiny$\bullet$}}
%% \put (5, 5) {\makebox(0,0){\tiny$\bullet$}}
%% \put (5, 25) {\vector (1, 0) {20}}
%% \put (5, 25) {\vector (1, -1) {20}}
%% \put (25, 25) {\vector (-1, -1) {20}}
%% \put (25, 5) {\vector (-1, 0) {20}}
%% \end{picture}
%% \begin{picture}(30,30)
%% \put (5, 25) {\makebox(0,0){\tiny$\bullet$}}
%% \put (25, 25) {\makebox(0,0){\tiny$\bullet$}}
%% \put (25, 5) {\makebox(0,0){\tiny$\bullet$}}
%% \put (5, 5) {\makebox(0,0){\tiny$\bullet$}}
%% \put (5, 25) {\vector (1, 0) {20}}
%% \put (5, 25) {\vector (0, -1) {20}}
%% \put (25, 5) {\vector (0, 1) {20}}
%% \put (25, 5) {\vector (-1, 0) {20}}
%% \end{picture}


\begin{acknowledgements}
The authors thank anonymous referees for helpful suggestions.
\end{acknowledgements}