\newpage\section{Levi Graph}\label{levi} %<––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––> %<–––––––––––––––––––– Levy –––––––––––––––––––––––––––––––> %<––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––> \begin{NewMacroBox}{grLevi}{\oarg{options}} \medskip From Wikipedia \url{http://en.wikipedia.org/wiki/Levi_graph} \emph{In combinatorics a Levi graph or incidence graph is a bipartite graph associated with an incidence structure. From a collection of points and lines in an incidence geometry or a projective configuration, we form a graph with one vertex per point, one vertex per line, and an edge for every incidence between a point and a line.\hfil\break In the mathematical field of graph theory, the Tutte–Coxeter graph or Tutte eight-cage is a 3-regular graph with 30 vertices and 45 edges. As the unique smallest cubic graph of girth 8 it is a cage and a Moore graph. It is bipartite, and can be constructed as the Levi graph of the generalized quadrangle. } From MathWord : \url{http://mathworld.wolfram.com/LeviGraph.html} \emph{It has 30 nodes and 45 edges. It has girth 8, diameter 4, chromatic number 2. The Levi graph is a generalized polygon which is the point/line incidence graph of the generalized quadrangle . The graph was first discovered by Tutte (1947), and is also called the Tutte-Coxeter graph , Tutte's cage or "Tutte's (3,8)-cage".The Levi graph is the unique (3,8)-cage graph.\hfil\break The incidence graph of a generic configuration is sometimes known as a Levi graph (Coxeter 1950).} \href{http://mathworld.wolfram.com/topics/GraphTheory.html}% {\textcolor{blue}{MathWorld}} by \href{http://en.wikipedia.org/wiki/Eric_W._Weisstein}% {\textcolor{blue}{E.Weisstein}} Some examples of Levi Graphs with this definition are~: \begin{itemize} \item Desargues graph \item Heawood graph \item Heawood graph \item Pappus graph \item Gray graph \item Tutte eight-cage \end{itemize} \end{NewMacroBox} The two forms can be draw with : \begin{tkzexample}[code only] \grLevi[RA=7]\end{tkzexample} and \begin{tkzexample}[code only] \grLevi[form=2,RA=7,RB=5,RC=3]\end{tkzexample} You can see on the next pages, the two forms. \vfill\newpage Now I show you how to code this graph. \subsection{\tkzname{Levy graph :form 1}} \bigskip \begin{center} \begin{tkzexample}[vbox] \begin{tikzpicture} \GraphInit[vstyle=Art] \grLCF[prefix=a,RA=6]{-13,-9,7,-7,9,13}{5} \end{tikzpicture} \end{tkzexample} \end{center} \vfill\newpage \subsection{\tkzname{Levy graph :form 2}} \bigskip \begin{center} \begin{tkzexample}[vbox] \begin{tikzpicture} \GraphInit[vstyle=Art] \grCycle[prefix=a,RA=7]{10} \EdgeInGraphMod{a}{10}{5} \grEmptyCycle[prefix=b,RA=5]{10} \grEmptyCycle[prefix=c,RA=3]{10} \EdgeInGraphMod{c}{10}{4} \end{tikzpicture} \end{tkzexample} \end{center} \endinput