Argthtjtr

I want to draw

But I don't know how to create v_6 which is the intersection of Edge(v_3)(v_5) & Edge(v_1)(v_4) and v_7 is the intersection of Edge(v_3)(v_5) & Edge(v_2)(v_4).
Here the code for the pentagram

documentclass{standalone}

usepackage{tikz,tkz-graph,tkz-berge}

%usetikzlibrary{positioning,fit,patterns} 



begin{document}

begin{tikzpicture}[rotate=90]

GraphInit[vstyle=Classic]

tikzset{VertexStyle/.append style={minimum size=3pt, inner sep=3pt}}

Vertices[Math,Lpos=90,unit=2]{circle}{v_1,v_2,v_3,v_4,v_5}

Edges(v_1,v_2,v_3,v_4,v_5,v_1)

Edges(v_1,v_3,v_5,v_2,v_4,v_1)

%WE[Math,Lpos=90,unit=2.75](v_1){v_7} % Brutal

%SOEA[Math,Lpos=0,unit=.5](v_7){v_6} 

end{tikzpicture}

end{document} 

Here is an alternative to JouleV's nice answer in which you do not have to redraw any line, i.e. can keep what you had.

documentclass{standalone}

usepackage{tikz,tkz-graph,tkz-berge}

usetikzlibrary{calc} 



begin{document}

begin{tikzpicture}[rotate=90]

GraphInit[vstyle=Classic]

tikzset{VertexStyle/.append style={minimum size=3pt, inner sep=3pt}}

Vertices[Math,Lpos=90,unit=2]{circle}{v_1,v_2,v_3,v_4,v_5}

Edges(v_1,v_2,v_3,v_4,v_5,v_1)

Edges(v_1,v_3,v_5,v_2,v_4,v_1)

%WE[Math,Lpos=90,unit=2.75](v_1){v_7} % Brutal

%SOEA[Math,Lpos=0,unit=.5](v_7){v_6} 

path  (intersection cs:first line={(v_3)--(v_5)}, second line={(v_1)--(v_4)})

node[circle,fill,label=right:$v_6$] (v_6){} 

(intersection cs:first line={(v_3)--(v_5)}, second line={(v_2)--(v_4)})

node[circle,fill,label=below:$v_7$] (v_7){} ;

end{tikzpicture}

end{document} 

I would do

documentclass[tikz]{standalone}

usetikzlibrary{shapes.geometric, intersections}

begin{document}

begin{tikzpicture}

    node[draw, minimum size = 3cm, regular polygon, regular polygon sides = 5] (a) {};

    foreach x in {1,2,...,5} 

        fill (a.corner x) circle [radius=2pt];

    draw (a.corner 1) node[above] {$v_1$};

    draw (a.corner 2) node[left] {$v_2$};

    draw (a.corner 3) node[below] {$v_3$};

    draw (a.corner 4) node[below] {$v_4$};

    draw (a.corner 5) node[right] {$v_5$};

    draw (a.corner 1)--(a.corner 3);

    draw (a.corner 2)--(a.corner 5);

    draw [name path = seg1] (a.corner 2)--(a.corner 4);

    draw [name path = seg2] (a.corner 1)--(a.corner 4);

    draw [name path = comm] (a.corner 3)--(a.corner 5);

    path [name intersections = {of = seg1 and comm, by = inter1}];

    fill (inter1) circle [radius = 2pt];

    draw (inter1) node[below] {$v_7$};

    path [name intersections = {of = seg2 and comm, by = inter2}];

    fill (inter2) circle [radius = 2pt];

    draw (inter2) node[below right] {$v_6$};

end{tikzpicture}

end{document}

documentclass[12pt]{standalone}

usepackage{pst-poly,pst-eucl} 

begin{document}

begin{pspicture}(-3,-2.5)(3,3)

psset{unit=2.5cm,PstPicture=false,dotsize=0.1}

PstStarFiveLines

providecommand{PstPolygonNode}{psdots(1;INode)}

PstPentagon[PolyName=A]

pstInterLL[PosAngle=-35]{A1}{A4}{A2}{A5}{v_6}

pstInterLL[PosAngle=-90]{A1}{A4}{A3}{A5}{v_7}

uput[90](A2){$v_1$}

uput[180](A3){$v_2$}

uput[-90](A4){$v_3$}

uput[-90](A5){$v_4$}

uput[0](A1){$v_5$}

end{pspicture}

end{document}