Construct a Graph with 7 Vertex, 21 Edges on 2D Plane On a 2D plane, how to construct a graph with 7 vertexes and 21 edges? I tried various combination but couldn't seem to draw that kind of contrived graph on a paper.
But my understanding is that it is possible. So anyone can help me with it?
 A: My guess is you're after $K_7$ drawn on the torus; see http://www.amotlpaa.org/math/k7torus.html
To answer you question as to why $K_5$ is not planar:  If a complete graph is planar, then for every $K_3$ subgraph, either every vertex (that's not part of the $K_3$) is inside the $K_3$ or outside the $K_3$ (otherwise there's a crossing edge from outside to inside the $K_3$).
So, if we attempt to draw $K_5$ vertex-by-vertex, then we first draw a triangle $K_3$.  We can place the 4-th vertex either inside or outside of the triangle.  In either case, the drawing obtained will look like:

Now, wherever you put the 5-th vertex, you will form some $K_3$ subgraph for which the other two vertices are not both inside and not both outside.
A: Although the software below does not allow one to check for planarity you might find it useful in seeing drawings of graphs (not typically drawn in the plane) with a small number of vertices and specified number of edges:
http://www.gfredericks.com/sandbox/graphs/browse 
