Just with the help of TikZ it is perfectly possible to do such a job. Here is indeed a possible solution that allows you to

- easily set the number of points on the circle;
- set the circle radius;
- decide the position of the picture in terms of coordinates
- eventually position labels.

The first three things are did by one command: `\drawconnectvertices`

while to add labels one should use `\drawconnectverticeslabelled`

.

Here is the code (revised thanks to Brent suggestion in the comments):

```
\documentclass{article}
\usepackage{tikz}
\usetikzlibrary{calc,shapes.geometric} % required for the polygon shape
\def\drawconnectvertices[num vertex=#1, circle radius=#2] at (#3);{%
\pgfmathtruncatemacro\vertices{#1}
\pgfmathsetmacro\circleradius{#2}
\pgfmathsetmacro\halfcircleradius{\circleradius/2}
\draw[blue] (#3) circle (\halfcircleradius cm) node[regular polygon, regular polygon sides=\vertices, minimum size=\circleradius cm, draw=none, name={vertex set}] {};
\foreach \x in {1,...,\vertices}{
\node[draw,circle, inner sep=1pt,blue, fill=blue] at (vertex set.corner \x) {};
}
\foreach \x in {1,...,\vertices}{
\foreach \y in {\x,...,\vertices}{
\draw[ultra thin, red] (vertex set.corner \x)--(vertex set.corner \y);
}
}
}
\def\drawconnectverticeslabelled[num vertex=#1, circle radius=#2, shift angle=#3] at (#4);{%
\pgfmathtruncatemacro\vertices{#1}
\pgfmathsetmacro\circleradius{#2}
\pgfmathsetmacro\halfcircleradius{\circleradius/2}
\draw[blue] (#4) circle (\halfcircleradius cm) node[regular polygon, regular polygon sides=\vertices, minimum size=\circleradius cm, draw=none, name={vertex set}] {};
\foreach \x in {1,...,\vertices}{
\node[draw,circle, inner sep=1pt,blue, fill=blue] at (vertex set.corner \x) {};
\pgfmathparse{#3-360*(\x-1)/ \vertices}
\node at ($(vertex set)+(\pgfmathresult:\halfcircleradius)$)[label={[font=\small]\pgfmathresult:$\x$}]{};
}
\foreach \x in {1,...,\vertices}{
\foreach \y in {\x,...,\vertices}{
\draw[ultra thin, red] (vertex set.corner \x)--(vertex set.corner \y);
}
}
}
\begin{document}
\begin{tikzpicture}
\drawconnectverticeslabelled[num vertex=6, circle radius=3, shift angle=0] at (0,0.75);
\drawconnectverticeslabelled[num vertex=4, circle radius=2, shift angle=45] at (4,0.75);
\drawconnectvertices[num vertex=8, circle radius=3] at (8,0.75);
\drawconnectverticeslabelled[num vertex=15, circle radius=6, shift angle=90] at (4,-5);
\drawconnectvertices[num vertex=30, circle radius=8] at (4,-13.5);
\end{tikzpicture}
\end{document}
```

The output:

One remark on `\drawconnectverticeslabelled`

: as it is possible to see from the example, the order of the labels could be changed according to the `shift angle`

.

After having seen TeX Parameter Processing imitating key-value pairs, here is a complete TikZ-based solution.

I've defined `\drawconnectvertices`

as an alias of `\node`

and now the parameters and the type of drawing are set up with `pgfkeys`

:

parameters:

`num vertex`

;
`circle radius`

;
`shift angle`

;
`at pos`

*(new)*: provides the coordinates in which the node is positioned;

type of drawing:

`circumference with points`

;
`circumference with points labelled`

.

Notice that the parameters should *always* be declared before the type of drawing.

The code:

```
\documentclass{article}
\usepackage[height=22cm]{geometry}
\usepackage{tikz}
\usetikzlibrary{calc,shapes.geometric} % required for the polygon shape
\pgfkeys{/tikz/.cd,
num vertex/.initial=6,
num vertex/.get=\vertices,
num vertex/.store in=\vertices,
circle radius/.initial=3,
circle radius/.get=\circleradius,
circle radius/.store in=\circleradius,
shift angle/.initial=0,
shift angle/.get=\shiftangle,
shift angle/.store in=\shiftangle,
at pos/.initial={(0,0)},
at pos/.get=\position,
at pos/.store in=\position,
}
% that's just an alias for \node
\makeatletter
\def\drawconnectvertices{\tikz@path@overlay{node}}
\makeatother
\pgfkeys{/tikz/circumference with points/.code={
\pgfmathsetmacro\halfcircleradius{\circleradius/2}
\draw[blue] \position circle (\halfcircleradius cm) node[regular polygon, regular polygon sides=\vertices, minimum size=\circleradius cm, draw=none, name={vertex set}] {};
\foreach \x in {1,...,\vertices}{
\node[draw,circle, inner sep=1pt,blue, fill=blue] at (vertex set.corner \x) {};
}
\foreach \x in {1,...,\vertices}{
\foreach \y in {\x,...,\vertices}{
\draw[ultra thin, red] (vertex set.corner \x)--(vertex set.corner \y);
}
}
}
}
\pgfkeys{/tikz/circumference with points labelled/.code={
\pgfmathsetmacro\halfcircleradius{\circleradius/2}
\draw[blue] \position circle (\halfcircleradius cm) node[regular polygon, regular polygon sides=\vertices, minimum size=\circleradius cm, draw=none, name={vertex set}] {};
\foreach \x in {1,...,\vertices}{
\node[draw,circle, inner sep=1pt,blue, fill=blue] at (vertex set.corner \x) {};
\pgfmathparse{\shiftangle-360*(\x-1)/ \vertices}
\node at ($(vertex set)+(\pgfmathresult:\halfcircleradius)$)[label={[font=\small]\pgfmathresult:$\x$}]{};
}
\foreach \x in {1,...,\vertices}{
\foreach \y in {\x,...,\vertices}{
\draw[ultra thin, red] (vertex set.corner \x)--(vertex set.corner \y);
}
}
}
}
\begin{document}
\begin{tikzpicture}
\drawconnectvertices[at pos={(0,0.75)}, circumference with points labelled] {};
\drawconnectvertices[num vertex=4,
circle radius=2,
at pos={(4,0.75)},
shift angle=45,circumference with points labelled] {};
\drawconnectvertices[num vertex=8,
at pos={(8,0.75)},
circumference with points] {};
\drawconnectvertices[num vertex=15,
circle radius=6,
shift angle=90,
at pos={(4,-5)},
circumference with points labelled] {};
\drawconnectvertices[num vertex=30,
circle radius=8,
at pos={(4,-13.5)},
circumference with points] {};
\end{tikzpicture}
\end{document}
```

provides the same result displayed above.

## Best Answer

After lot's of pain, I finally came up with this solution. If you find a better/shorter way to proceed, at any point, please let me now !

Output: