%!TEX root = /Users/ego/Boulot/TKZ/tkz-berge/NamedGraphs/doc/NamedGraphs-main.tex \newpage\section{ The five Platonics Graphs} %<––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––> %<–––––––––––––––––––– Platonic graphs –––––––––––––––––––––––––––––––> %<––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––> The Platonic Graphs are the graphs formed by the edges and vertices of the five regular Platonic solids. The five Platonics Graphs are illustrated below. \begin{enumerate} \item tetrahedral \item octahedral \item cube \item icosahedral \item dodecahedral \end{enumerate} %<––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––> %<––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––> \begin{NewMacroBox}{grTetrahedral}{\oarg{RA=Number}} From MathWord : \url{http://mathworld.wolfram.com/TetrahedralGraph.html} \emph{\tkzname{Tetrahedral Graph} is the unique polyhedral graph on four nodes which is also the complete graph and therefore also the wheel graph . It is implemented as \tkzcname{grTetrahedral}} \href{http://mathworld.wolfram.com/TetrahedralGraph.html}% {\textcolor{blue}{MathWorld}} by \href{http://en.wikipedia.org/wiki/Eric_W._Weisstein}% {\textcolor{blue}{E.Weisstein} } It has : \begin{enumerate} \item 4 nodes, \item 6 edges, \item graph diameter 1. \end{enumerate} The Tetrahedral Graph is 3-Regular \end{NewMacroBox} \subsection{\tkzname{Tetrahedral}} \begin{center} \begin{tkzexample}[vbox] \begin{tikzpicture}[scale=.6] \GraphInit[vstyle=Shade] \renewcommand*{\VertexInnerSep}{4pt} \SetVertexNoLabel\SetGraphShadeColor{red!50}{black}{red} \grTetrahedral[RA=5] \end{tikzpicture} \end{tkzexample} \end{center} \clearpage\newpage \subsection{\tkzname{Tetrahedral LCF embedding}} \vspace*{2cm} \begin{center} \begin{tkzexample}[vbox] \begin{tikzpicture}[rotate=18] \renewcommand*{\VertexInnerSep}{8pt} \GraphInit[vstyle=Art] \SetGraphArtColor{red!50}{orange} \grLCF[RA=7]{2,-2}{2} \end{tikzpicture} \end{tkzexample} \end{center} \clearpage\newpage %<––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––> %<––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––> \begin{NewMacroBox}{grOctahedral}{\oarg{RA=\meta{Number},RB=\meta{Number}}} \medskip From MathWord : \url{http://mathworld.wolfram.com/OctahedralGraph.html} \emph{\tkzname{Octahedral Graph} is isomorphic to the circulant graph $CI_{[1,2]}(6)$ . Two embeddings of this graph are illustrated below. It is implemented as \tkzcname{grOctahedral} or as \tkzcname{grSQCycle\{6\}}.} \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}} It has : \begin{enumerate} \item 6 nodes, \item 12 edges, \item graph diameter 2. \end{enumerate} \medskip The Octahedral Graph is 4-Regular. \end{NewMacroBox} \medskip \subsection{\tkzname{Octahedral}} \begin{center} \begin{tkzexample}[vbox] \begin{tikzpicture} \grOctahedral[RA=6,RB=2] \end{tikzpicture} \end{tkzexample} \end{center} \vfill\newpage\null \begin{center} \begin{tkzexample}[vbox] \begin{tikzpicture} \grSQCycle[RA=5]{6} \end{tikzpicture} \end{tkzexample} \end{center} \vfill\newpage\null %<––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––> %<––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––> \medskip \begin{NewMacroBox}{grCubicalGraph}{\oarg{RA=\meta{Number},RB=\meta{Number}}} \medskip From MathWord : \url{http://mathworld.wolfram.com/CubicalGraph.html} \emph{\tkzname{Cubical Graph} is isomorphic to a generalized Petersen graph $PG_{[4,1]}$, to a bipartite Kneser graph , to a crown graph and it is equivalent to the Cycle Ladder $CL(4)$. Two embeddings of this graph are illustrated below. It is implemented as \tkzcname{grCubicalGraph} or \tkzcname{grPrism\{4\}}.} \href{http://mathworld.wolfram.com/CubicalGraph.html}% {\textcolor{blue}{MathWorld}} by \href{http://en.wikipedia.org/wiki/Eric_W._Weisstein}% {\textcolor{blue}{E.Weisstein}} It has : \begin{enumerate} \item 8 nodes, \item 12 edges, \item graph diameter 3. \end{enumerate} The Cubical Graph is 3-Regular. \end{NewMacroBox} \subsection{\tkzname{Cubical Graph : form 1}} \begin{center} \begin{tkzexample}[vbox] \begin{tikzpicture} \grCubicalGraph[RA=5,RB=2] \end{tikzpicture} \end{tkzexample} \end{center} \vfill\newpage\null \subsection{\tkzname{Cubical Graph : form 2}} \begin{center} \begin{tkzexample}[vbox] \begin{tikzpicture} \grCubicalGraph[form=2,RA=7,RB=4] \end{tikzpicture} \end{tkzexample} \end{center} \vfill\newpage \subsection{\tkzname{Cubical LCF embedding}} \vspace*{2cm} \begin{center} \begin{tkzexample}[vbox] \begin{tikzpicture}[rotate=18] \GraphInit[vstyle=Art]\renewcommand*{\VertexInnerSep}{8pt} \SetGraphArtColor{red!50}{orange} \grLCF[RA=7]{3,-3}{4} \end{tikzpicture} \end{tkzexample} \end{center} \clearpage\newpage %<––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––> \begin{NewMacroBox}{grIcosahedral}{\oarg{RA=\meta{Number},RB=\meta{Number},RC=\meta{Number}}} \medskip From MathWord : \url{http://mathworld.wolfram.com/IcosahedralGraph.html} \emph{The \tkzname{Icosahedral Graph} is the Platonic graph whose nodes have the connectivity of the icosahedron, illustrated above in a number of embeddings. The icosahedral graph has 12 vertices and 30 edges. Since the icosahedral graph is regular and Hamiltonian, it has a generalized LCF notation.} \href{http://mathworld.wolfram.com/IcosahedralGraph.html}% {\textcolor{blue}{MathWorld}} by \href{http://en.wikipedia.org/wiki/Eric_W._Weisstein}% {\textcolor{blue}{E.Weisstein}} \medskip It has : \begin{enumerate} \item 12 nodes, \item 30 edges, \item graph diameter 3. \end{enumerate} \medskip The Icosahedral Graph is 5-Regular. \end{NewMacroBox} \medskip \subsection{\tkzname{Icosahedral forme 1 }} \tikzstyle{EdgeStyle}= [thick,% double = orange,% double distance = 1pt] \begin{center} \begin{tkzexample}[vbox] \begin{tikzpicture}[scale=.8] \GraphInit[vstyle=Art]\renewcommand*{\VertexInnerSep}{4pt} \SetGraphArtColor{red}{orange} \grIcosahedral[RA=5,RB=1] \end{tikzpicture} \end{tkzexample} \end{center} \clearpage\newpage \subsection{\tkzname{Icosahedral forme 2 }} \vspace*{2cm} \begin{center} \begin{tkzexample}[vbox] \begin{tikzpicture}[rotate=-30] \GraphInit[vstyle=Art] \renewcommand*{\VertexInnerSep}{8pt} \SetGraphArtColor{red!50}{orange} \grIcosahedral[form=2,RA=8,RB=2,RC=.8] \end{tikzpicture} \end{tkzexample} \end{center} \vfill\newpage \subsection{\tkzname{Icosahedral} \tkzname{RA=1} et \tkzname{RB=7}} \begin{center} \begin{tkzexample}[vbox] \begin{tikzpicture} \GraphInit[vstyle=Art] \renewcommand*{\VertexInnerSep}{8pt} \SetGraphArtColor{red!50}{orange} \grIcosahedral[RA=1,RB=7] \end{tikzpicture} \end{tkzexample} \end{center} \clearpage\newpage \subsection{\tkzname{Icosahedral LCF embedding 1}} \vspace*{2cm} \begin{center} \begin{tkzexample}[vbox] \begin{tikzpicture}[rotate=18] \GraphInit[vstyle=Art] \renewcommand*{\VertexInnerSep}{8pt} \SetGraphArtColor{red!50}{orange} \grLCF[RA=7]{-4,-3,4}{6} \end{tikzpicture} \end{tkzexample} \end{center} \clearpage\newpage \subsection{\tkzname{Icosahedral LCF embedding 2}} \vspace*{2cm} \begin{center} \begin{tkzexample}[vbox] \begin{tikzpicture}[rotate=18] \GraphInit[vstyle=Art] \SetGraphArtColor{red!50}{orange} \grLCF[RA=7]{-2,2,3}{6} \end{tikzpicture} \end{tkzexample} \end{center} \clearpage\newpage %<––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––> \begin{NewMacroBox}{grDodecahedral}{\oarg{RA=\meta{Number},RB=\meta{Number},RC=\meta{Number},RD=\meta{Number}}} \medskip From MathWord : \url{http://mathworld.wolfram.com/DodecahedralGraph.html} \emph{The \tkzname{Icosahedral Graph} is the Platonic graph corresponding to the connectivity of the vertices of a dodecahedron, illustrated above in four embeddings. The left embedding shows a stereographic projection of the dodecahedron, the second an orthographic projection, the third is from Read and Wilson, and the fourth is derived from LCF notation.} \href{http://mathworld.wolfram.com/DodecahedralGraph.html}% {\textcolor{blue}{MathWorld}} by \href{http://en.wikipedia.org/wiki/Eric_W._Weisstein}% {\textcolor{blue}{E.Weisstein}} \medskip It has : \begin{enumerate} \item 20 nodes, \item 30 edges, \item graph diameter 5. \end{enumerate} \medskip The Dodecahedral Graph is 3-Regular. \end{NewMacroBox} \medskip \subsection{\tkzname{Dodecahedral}} \begin{center} \begin{tkzexample}[vbox] \begin{tikzpicture}[rotate=18,scale=.6] \GraphInit[vstyle=Art] \SetGraphArtColor{red!50}{orange} \grDodecahedral[RA=7,RB=4,RC=2,RD=1] \end{tikzpicture} \end{tkzexample} \end{center} \subsection{\tkzname{Dodecahedral other embedding}} \vspace*{2cm} \begin{center} \begin{tkzexample}[vbox] \begin{tikzpicture} \grCycle[RA=7,prefix=a]{10} \grSQCycle[RA=4,prefix=b]{10} \foreach \v in {0,...,9} {\Edge(a\v)(b\v)} \end{tikzpicture} \end{tkzexample} \end{center} \vfill\newpage \subsection{\tkzname{Dodecahedral LCF embedding}} \vspace*{2cm} \begin{center} \begin{tkzexample}[vbox] \begin{tikzpicture}[rotate=18] \GraphInit[vstyle=Art] \SetGraphArtColor{red!50}{orange} \grLCF[RA=7]{10,7,4,-4,-7,10,-4,7,-7,4}{2} \end{tikzpicture} \end{tkzexample} \end{center} \endinput