\documentclass[a4paper,12pt]{report} 
\usepackage{tikz,amsmath}
\usepackage[fleqn]{amsmath}
\tikzstyle{vertex}=[circle, draw, inner sep=0pt, minimum size=3pt] 
\newcommand{\vertex}{\node[vertex]}
\begin{document}


\begin{center}

\end{center}



\begin{tikzpicture} [scale=2,very thick]
\vertex[fill] (v1) at (0,0) [label=left:$v_{1}$] {};
\vertex[fill] (v2) at (1,0) [label=above:$v_{2}$] {};
\vertex[fill] (v3) at (2,0) [label=right:$v_{3}$] {};
\vertex[fill] (v4) at (1.5,1) [label=right:$v_{4}$] {};
\vertex[fill] (v5) at (.5,1) [label=left:$v_{5}$] {};
\vertex[fill] (v6) at (.5,-1) [label=above:$v_{6}$] {};
\vertex[fill] (v7) at (1.5,-1) [label=right:$v_{7}$] {};
\draw (0,0) .. controls (.2,.5) .. (.5,1);
\draw (.5,1) .. controls (1,1.1) .. (1.5,1);
\draw (1.5,1) .. controls (1.8,.5) .. (2,0);


\draw (2,0) .. controls (1.8,-.5) .. (1.5,-1);
\draw (0,0) .. controls (.2,-.5) .. (.5,-1);
\draw (.5,-1) .. controls (1,-1.1) .. (1.5,-1);
\path
(v1) edge (v2)
(v4) edge (v2)
(v3) edge (v3)
(v6) edge (v2)
(v5) edge (v2)
(v7) edge (v2)
(v2) edge (v3);
\end{tikzpicture}  



\end{document}