\newpage\section{Hypercube} %<––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––> %<–––––––––––––––––––– Hypercube –––––––––––––––––––––––––––––––> %<––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––> From Wikipedia :\url{http://en.wikipedia.org/wiki/Hypercube_graph} In the mathematical field of graph theory, the hypercube graph $Q_n$ is a special regular graph with $2n$ vertices, which correspond to the subsets of a set with $n$ elements. Two vertices labelled by subsets S and T are joined by an edge if and only if S can be obtained from T by adding or removing a single element. Each vertex of $Q_n$ is incident to exactly $n$ edges (that is, $Q_n$ is $n$-regular), so the total number of edges is $2^{n-1}n$. The name comes from the fact that the hypercube graph is the one-dimensional skeleton of the geometric hypercube. Hypercube graphs should not be confused with cubic graphs, which are graphs that are 3-regular. The only hypercube that is a cubic graph is $Q_3$. \tikzstyle{VertexStyle} = [shape = circle,% fill = red,% inner sep = 3pt,% outer sep = 0pt,% draw] \SetVertexNoLabel \subsection{\tkzname{The hypercube graph $Q_4$} } The code is on the next page. \begin{center} \begin{tkzexample}[vbox] \begin{tikzpicture}[scale=.75] \grCycle[RA=8]{8} \pgfmathparse{8*(1-4*sin(22.5)*sin(22.5))} \let\tkzbradius\pgfmathresult \grCirculant[prefix=b,RA=\tkzbradius]{8}{3} \makeatletter \foreach \vx in {0,...,7}{% \pgfmathsetcounter{tkz@gr@n}{mod(\vx+1,8)} \pgfmathsetcounter{tkz@gr@a}{mod(\vx+7,8)} \pgfmathsetcounter{tkz@gr@b}{mod(\thetkz@gr@n+1,8)} \Edge(a\thetkz@gr@n)(b\thetkz@gr@b) \Edge(b\thetkz@gr@a)(a\vx) } \makeatother \end{tikzpicture} \end{tkzexample} \end{center} \endinput