The Hoffman-Singleton graph

\documentclass{article}

\pagestyle{empty}
\usepackage{tikz}

\begin{document}

\begin{center}
\begin{tikzpicture}[style=thick,scale=0.5]
\def\pentagon{(18:1cm) circle (4pt) node[above right=-1.75pt]{\tiny $4$} --
(90:1cm) circle (4pt) node[above]{\tiny $0$} --
(162:1cm) circle (4pt) node[above left=-1.75pt]{\tiny $1$} --
(234:1cm) circle (4pt) node[below]{\tiny $2$} --
(306:1cm) circle (4pt) node[below]{\tiny $3$} -- (18:1cm)}
\def\pentagram{(18:1cm) circle (4pt) node[above right=-1.75pt]{\tiny $4$} --
(162:1cm) circle (4pt) node[above left=-1.75pt]{\tiny $1$} --
(306:1cm) circle (4pt) node[below]{\tiny $3$} --
(90:1cm) circle (4pt) node[above]{\tiny $0$} --
(234:1cm) circle (4pt) node[below]{\tiny $2$} -- (18:1cm)}
\draw[yshift=4cm] node{\tiny $P_{0}$} \pentagon;
\draw[xshift=3cm,yshift=4cm] node{\tiny $P_{1}$} \pentagon;
\draw[xshift=6cm,yshift=4cm] node{\tiny $P_{2}$} \pentagon;
\draw[xshift=9cm,yshift=4cm] node{\tiny $P_{3}$} \pentagon;
\draw[xshift=12cm,yshift=4cm] node{\tiny $P_{4}$} \pentagon;
\draw node{\tiny $Q_{0}$} \pentagram;
\draw[xshift=3cm] node{\tiny $Q_{1}$} \pentagram;
\draw[xshift=6cm] node{\tiny $Q_{2}$} \pentagram;
\draw[xshift=9cm] node{\tiny $Q_{3}$} \pentagram;
\draw[xshift=12cm] node{\tiny $Q_{4}$} \pentagram;
\draw (6,-2.5) node%
{Vertex $i$ in $P_{j}$ is joined to vertex $i+jk\!\!\!\!\pmod{5}$ of $Q_{k}$};
\end{tikzpicture}
\end{center}
\end{document}

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