%<–––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––>
\section{Macros and Edges in a graph}
%<–––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––>
\subsection{Edge in a graph from one vertex \tkzcname{EdgeInGraphFromOneToComp}}

\begin{NewMacroBox}{EdgeInGraphFromOneToComp}{\oarg{local options}\var{prefix}\var{order}\var{from}}

\begin{tabular}{llc}
\hline
Arguments   &   & Definition              \\
\midrule
\TAline{order} {}{order of the graph}
\bottomrule
\end{tabular}

\medskip
\begin{tabular}{llc}
\midrule
options   & default  & definition                                           \\
\midrule
\TOline{RA}     {4}      { radius  circle}
\TOline{prefix} {a}      {prefix for vertices }
\TOline{Math}   {false}  {math mode }
\bottomrule
\end{tabular}

\medskip
\emph{This macro works on an unique graph. |from| is integer. |EdgeInGraph| designs a macro that works only in a graph defined by a prefix. The result is some edges between the vertex |from| and the others vertices. }
\end{NewMacroBox}





\subsubsection{Empty Cycle}
\begin{center}
\begin{tkzexample}[very small]
\begin{tikzpicture}
  \GraphInit[vstyle=Shade]
  \grEmptyCycle[RA=4,prefix=a]{8}%
  \EdgeInGraphFromOneToComp{a}{8}{3}
\end{tikzpicture}
\end{tkzexample}
\end{center}
 
\vfill
\newpage
\subsection{Edges in a graph - a loop \tkzcname{EdgeInGraphLoop}}%
\begin{NewMacroBox}{EdgeInGraphLoop}{\var{prefix}\var{order}}
\emph{This macro is useful with vertices on a circle . |order| in an integer.}
\end{NewMacroBox}


\subsubsection{Empty Cycle}
\begin{center}
  \begin{tkzexample}[very small]
  \begin{tikzpicture}
  \GraphInit[vstyle=Shade]
  \grEmptyCycle[RA=2,prefix=a]{8}%
  \EdgeInGraphLoop{a}{8}
  \end{tikzpicture}\end{tkzexample}
\end{center}

\subsubsection{Empty Cycle}
\begin{center}
\begin{tkzexample}[very small]
\begin{tikzpicture}[node distance=4cm]
 \GraphInit[vstyle=Shade]
 \Vertices{square}{a0,a1,a2,a3}
 \EdgeInGraphLoop{a}{4}
\end{tikzpicture}
\end{tkzexample}
\end{center}

\newpage
\subsection{Edges in a graph - a loop \tkzcname{EdgeInGraphLoop*}}
\begin{NewMacroBox}{EdgeInGraphLoop*}{\var{prefix}\var{order}}

\medskip
\emph{Not exactly a loop, there is no edge between the first and the last vertex.}
\end{NewMacroBox}

\subsubsection{Empty Cycle}
\begin{center}
\begin{tkzexample}[very small]
\begin{tikzpicture}
  \GraphInit[vstyle=Art]
  \grEmptyCycle[RA=4,prefix=a]{8}%
  \EdgeInGraphLoop*{a}{8}
\end{tikzpicture}
\end{tkzexample}
\end{center}

\subsubsection{Empty Path}
\begin{center}
\begin{tkzexample}[very small]
\begin{tikzpicture}
     \grEmptyPath[prefix=h,RA=2,RS=2]{6}
     \EdgeInGraphLoop*{h}{6}
\end{tikzpicture}
\end{tkzexample}
\end{center}

\vfill
\newpage
\subsection{Sequence of edges in a graph  \tkzcname{EdgeInGraphSeq}}
\begin{NewMacroBox}{EdgeInGraphSeq}{\var{prefix}\var{start}\var{end}}

\medskip
\emph{This macro gives a sequence of edges between |start| and |end|.\\
|start| and |end| are  two integers. }
\end{NewMacroBox}

\subsubsection{EdgeInGraphSeq}
\begin{center}
\begin{tkzexample}[very small]
\begin{tikzpicture}
  \GraphInit[vstyle=Shade]
  \grEmptyCycle[RA=2,prefix=a]{8}%
  \EdgeInGraphSeq{a}{2}{5}
\end{tikzpicture}
\end{tkzexample}
\end{center}

\newpage
\subsection{Edges in a graph  \tkzcname{EdgeInGraphMod}}
\begin{NewMacroBox}{EdgeInGraphMod}{\var{prefix}\var{order}\var{add}}

\medskip
\emph{This macro works on an unique graph. Edges between $v_i$ and $v_j$ with $i$ in $0,...,(\text{\#2}-1)$  and $j=\text{Mod(i+\#3,\#2)}$.\\
\#2 = |order| and  \#3 = |add|.\\
|Mod| is like |mod| but the result is a positive integer. }
\end{NewMacroBox}

\subsubsection{EdgeInGraphMod}
\begin{center}
\begin{tkzexample}[very small]
\begin{tikzpicture}
  \GraphInit[vstyle=Shade]
  \grEmptyCycle[RA=2,prefix=a]{8}%
  \EdgeInGraphMod{a}{8}{2}
\end{tikzpicture}
\end{tkzexample}
\end{center}

\subsubsection{EdgeInGraphMod 2}
\begin{center}
\begin{tkzexample}[very small]
\begin{tikzpicture}
  \GraphInit[vstyle=Shade]
  \grEmptyCycle[RA=3,prefix=a]{13}%
  \EdgeInGraphMod{a}{13}{3}
\end{tikzpicture}
\end{tkzexample}
\end{center}

\newpage
\subsection{Edges in a graph  \tkzcname{EdgeInGraphMod*}}
\begin{NewMacroBox}{EdgeInGraphMod*}{\var{prefix}\var{order}\var{add}\var{start}\var{step}}

\medskip
\emph{Edges between $v_i$ and $v_j$ with $i$ in $\#4,\#4+\#5,...,(\text{\#2}-1)$  and $j=\text{Mod(i+\#3,\#2)}$}\\
\#2 = |order|,  \#3 = |add|, \#4 = |start|, \#5 = |step|.\\
\end{NewMacroBox}

\subsubsection{EdgeInGraphMod*}
\begin{center}
\begin{tkzexample}[very small]
\begin{tikzpicture}
  \GraphInit[vstyle=Shade]
  \grEmptyCycle[prefix=a]{17}%
  \EdgeInGraphMod*{a}{17}{5}{1}{2}
\end{tikzpicture}
\end{tkzexample}
\end{center}

\newpage
\subsection{Edges in a graph  \tkzcname{EdgeInGraphModLoop}}
\begin{NewMacroBox}{EdgeInGraphModLoop}{\var{prefix}\var{order}\var{add}\var{start}}

\medskip
\emph{ |order|, |add| and |start| are integers.\\
Edges between $v_i$ and $v_j$ with $i$ from $\#4$, $j=\text{Mod(i+\#3,\#2)}$ and then $i=j$ until $j=\#4$\\
\#2 = |order|,  \#3 = |add| and \#4 = |start|.}
\end{NewMacroBox}

\subsubsection{EdgeInGraphModLoop}
\begin{center}
\begin{tkzexample}[very small]
\begin{tikzpicture}
  \GraphInit[vstyle=Shade]
  \grEmptyCycle[RA=4]{7}
  \EdgeInGraphModLoop{a}{7}{2}{1}
\end{tikzpicture}
\end{tkzexample}
\end{center}

\subsubsection{EdgeInGraphModLoop}
\begin{center}
\begin{tkzexample}[very small]
\begin{tikzpicture}
  \GraphInit[vstyle=Shade]
  \grEmptyCycle[RA=4]{8}
  \EdgeInGraphModLoop{a}{8}{2}{1}
\end{tikzpicture}
\end{tkzexample}
\end{center}


\newpage
\subsection{Edges between two graphs with the same order \tkzcname{EdgeIdentity}}

\begin{NewMacroBox}{EdgeIdentity}{\var{prefix1}\var{prefix2}\var{order}}

\medskip
\emph{|order| is an integer. This macro gives edges between two graphs.\\
Edges between $v_i$ and $v_j$ with $i=j$ in $0,...,(\text{\#3}-1)$.\\
\#3 = |order|.\\}
\end{NewMacroBox}

\subsubsection{EdgeIdentity}
\begin{center}
\begin{tkzexample}[very small]
\begin{tikzpicture}
  \GraphInit[vstyle=Shade]
    \grEmptyCycle[prefix=v,RA=3]{5}
    \grEmptyCycle[prefix=w,RA=1]{5}
    \EdgeIdentity{v}{w}{5}
\end{tikzpicture}
\end{tkzexample}
\end{center}

\vfill
\newpage
\subsection{Edges between two graphs with the same order \tkzcname{EdgeIdentity*}}

\begin{NewMacroBox}{EdgeIdentity*}{\var{prefix1}\var{prefix2}\var{list}}

\medskip
\emph{|list| is a list of  integers. This macro gives edges between two graphs.\\
Edges between $v_i$ and $v_j$ with $i=j$ in |list|.\\}

\end{NewMacroBox}

\subsubsection{EdgeIdentity*}
\begin{center}
\begin{tkzexample}[very small]
\begin{tikzpicture}
  \GraphInit[vstyle=Shade]
  \begin{scope}[rotate=30]
  \grEmptyCycle[RA=3,prefix=a]{5}%
  \end{scope}
  \grEmptyCycle[RA=5,prefix=b]{5}%
   \EdgeIdentity*{a}{b}{0,...,4}
\end{tikzpicture}
\end{tkzexample}
\end{center}

\subsubsection{EdgeIdentity*}
\begin{center}
\begin{tkzexample}[very small]
\begin{tikzpicture}
 \GraphInit[vstyle=Shade]
  \begin{scope}[rotate=30]
  \grEmptyCycle[RA=3,prefix=a]{5}%
  \end{scope}
  \grEmptyCycle[RA=5,prefix=b]{5}%
   \EdgeIdentity*{a}{b}{0,2,4}
\end{tikzpicture}
\end{tkzexample}
\end{center}

\newpage
\subsection{Edges between two graphs \tkzcname{EdgeFromOneToAll}}
\begin{NewMacroBox}{EdgeFromOneToAll}{\var{prefix1}\var{prefix2}\var{from}\var{order}}

\medskip
\emph{The graphs must to have the same order. |from| and |order| are integers.}
\end{NewMacroBox}

\subsubsection{EdgeFromOneToAll}
\begin{center}
\begin{tkzexample}[very small]
\begin{tikzpicture}
  \GraphInit[vstyle=Shade]
  \grPath[form=1,RA=2,RS=0]{5}
  \grPath[form=1,prefix=b,RA=2,RS=4]{5}
  \EdgeFromOneToAll{a}{b}{1}{5}
\end{tikzpicture}\end{tkzexample}
\end{center}

\newpage
\subsection{Edges between two graphs \tkzcname{EdgeFromOneToSeq}}
\begin{NewMacroBox}{EdgeFromOneToSeq}{\var{prefix1}\var{prefix2}\var{from}\var{start}\var{end}}

\medskip
\emph{|from|, |start| and |end| are integers. This macro  builds edges between the vertex with an indice |from| through the vertices with an indice in the sequence |start|,...,|end|.}
\end{NewMacroBox}

\subsubsection{EdgeFromOneToSeq}
\begin{center}
\begin{tkzexample}[very small]
\begin{tikzpicture}
  \GraphInit[vstyle=Shade]
  \grPath[form=1,RA=2,RS=0]{5}
  \grPath[form=1,prefix=b,RA=2,RS=4]{5}
  \EdgeFromOneToSeq{a}{b}{1}{2}{4}
\end{tikzpicture}
\end{tkzexample}
\end{center}

\newpage
\subsection{Edges between two graphs \tkzcname{EdgeFromOneToSel}}
\begin{NewMacroBox}{EdgeFromOneToSel}{\var{prefix1}\var{prefix2}\var{from}\var{list}}

\medskip
\emph{This macro  builds edges between the vertex with an indice |from| through the vertices with an indice in the list |list|.}
\end{NewMacroBox}


\subsubsection{EdgeFromOneToSel}
\begin{center}
\begin{tkzexample}[very small]
\begin{tikzpicture}
  \GraphInit[vstyle=Shade]
  \grPath[form=1,RA=2]{5}
  \grPath[form=1,prefix=b,RA=2,RS=4]{5}
  \EdgeFromOneToSel{a}{b}{1}{0,2,4}
\end{tikzpicture}
\end{tkzexample}
\end{center}

\newpage
\subsection{Edges between two graphs \tkzcname{EdgeFromOneToComp}}
\begin{NewMacroBox}{EdgeFromOneToComp}{\var{prefix1}\var{prefix2}\var{from}\var{order2}}

\medskip
\emph{This macro  builds edges between the vertex with an indice |from| through all the vertices of the second graph, except the vertex with an indice |from|.}
\end{NewMacroBox}

\subsubsection{EdgeFromOneToComp}
\begin{center}
\begin{tkzexample}[vbox]
\begin{tikzpicture}
  \GraphInit[vstyle=Shade]
  \grPath[form=1,RA=2,RS=0]{5}
  \grPath[form=1,prefix=b,RA=2,RS=4]{5}
  \EdgeFromOneToComp{a}{b}{2}{3}
\end{tikzpicture}
\end{tkzexample}
\end{center}

\newpage
\subsection{Edges between two graphs \tkzcname{EdgeMod}}%
\begin{NewMacroBox}{EdgeMod}{\var{prefix1}\var{prefix2}\var{order}\var{step}}

\medskip
\emph{This macro works on two graphs with the same order. We get edges between $v_i$ and $v_j$ with $i$ in $0,...,(\text{\#2}-1)$  and $j=\text{Mod(i+\#4,\#3)}$.\\
\#3 = |order| and  \#4 = |step|.}
\end{NewMacroBox}

\subsubsection{EdgeMod}
\begin{center}
\begin{tkzexample}[vbox]
\begin{tikzpicture}
 \GraphInit[vstyle=Shade]
 \grEmptyCycle[prefix=a,RA=6]{9}
 \grEmptyCycle[prefix=b,RA=3]{9}
 \EdgeMod{a}{b}{9}{1}
\end{tikzpicture}
\end{tkzexample}
\end{center}

\newpage
\subsection{Edges between two graphs \tkzcname{EdgeMod*}}%
\begin{NewMacroBox}{EdgeMod*}{\var{prefix1}\var{prefix2}\var{order}\var{step1}\var{step2}}

\medskip
\emph{This macro works on two graphs with the same order. We get edges between $v_i$ and $v_j$ with $i$ in $0,...,(\text{\#3}-1)$ with a step $\text{\#5}$ and $j=\text{Mod(i+\#4,\#3)}$.\\
\#3 = |order| ,  \#4 = |step1| and \#5 = |step2|.}
\end{NewMacroBox}


\subsubsection{\tkzcname{EdgeMod*} }%with |step1|=1 and |step2|=2
\begin{center}
\begin{tkzexample}[vbox]
\begin{tikzpicture}
 \GraphInit[vstyle=Shade]
 \grEmptyCycle[prefix=a,RA=6]{8}
 \grEmptyCycle[prefix=b,RA=4]{8}
 \EdgeMod*{a}{b}{8}{1}{2}
\end{tikzpicture}
\end{tkzexample}
\end{center}


\subsubsection{EdgeMod* }%with |step1|=2 and |step2|=1
\begin{center}
\begin{tkzexample}[vbox]
\begin{tikzpicture}
 \GraphInit[vstyle=Shade]
 \grEmptyCycle[prefix=a,RA=6]{8}
 \grEmptyCycle[prefix=b,RA=2]{8}
 \EdgeMod*{a}{b}{8}{1}{1}
\end{tikzpicture}
\end{tkzexample}
\end{center}

\newpage
\subsection{Edges between two graphs \tkzcname{EdgeDoubleMod}}%
\begin{NewMacroBox}{EdgeDoubleMod}{\var{prefix1}\var{nb}\var{nb}\var{nb}\var{prefix2}\var{nb}\var{nb}\var{nb}\var{end}}

For the first node,  the numbers are :
\var{order1}\var{start1}\var{add1}

\medskip
For the second node,  the numbers are :
\var{order2}\var{start2}\var{add2}\var{end}

\medskip
\emph{Edges between $v_i$ and $v_j$ with $i=\text{Mod(\#3+(\#4*k),\#2)}$ and j=$\text{Mod(\#7+(\#8*k),\#6)}$ $k$ is an integer from $0$ to |end|.\\
\#2 = |order1|,  \#3 = |start1| and \#4 = |add1|.\\
\#6 = |order2|,  \#7 = |start2| and \#8 = |add2|.}
\end{NewMacroBox}


\subsubsection{EdgeDoubleMod}
\begin{center}
\begin{tkzexample}[vbox]
\begin{tikzpicture}
  \GraphInit[vstyle=Shade]
    \begin{scope}[rotate=-90]
      \grEmptyCycle[RA=2,prefix=a]{5}
    \end{scope}
    \begin{scope}[rotate=-18]
      \grEmptyCycle[RA=4,prefix=b]{5}
    \end{scope}
    \EdgeDoubleMod{b}{5}{0}{1}%
                 {a}{5}{2}{1}{5}
\end{tikzpicture}
\end{tkzexample}
\end{center}


\subsubsection{EdgeDoubleMod with two graphs and different orders}
\begin{center}
\begin{tkzexample}[vbox]
\begin{tikzpicture}
  \GraphInit[vstyle=Shade]
  \grEmptyCycle[prefix=a,RA=5]{10}
  \grEmptyCycle[prefix=b,RA=7]{20}
   \EdgeDoubleMod{a}{10}{0}{1}%
                 {b}{20}{0}{2}{10}
\end{tikzpicture}
\end{tkzexample}
\end{center}
\endinput
