1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
|
\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
|