A couple of facts about the Petersen graph (depicted somewhat poorly below):
It is nonhamiltonian. That is, there is no cycle within it that contains every vertex (also known as a Hamilton cycle). It is not a planar graph. That is, there is no way to move the vertices and edges around (keeping them attatched!) so that none of the edges intersect.

```
The vertices are designated by the shapes
made of #'s, and the edges are the lines between them.

#
__###__
__/   #   \__
__/      |      \__
__/         |         \__
__/            #            \__
__/              ###              \__
__/                / # \                \__
# __/                   |   |                   \__ #
###,__                  /     \                  __,###
#    \__#              |     |              #__/    #
\      ###------------/-------\------------###     /
|      #\_           |       |           _/#      |
\         \_        /         \        _/        /
|          \_      |         |      _/          |
\            \__  /           \  __/           /
|              \_|           |_/              |
\               /\_         _/\              /
|              |  \_     _/  |              |
\             /     \___/     \            /
|            |     _/ \_     |            |
\           /    _/     \_    \          /
|          |  _/         \_  |          |
\         # _/             \_ #        /
|       ###                 ###       |
\     _/ #                   # \_    /
|   /                           \   |
\#/                             \#/
###-----------------------------###
#                               #
```

I'm open to any suggestions--about how to improve this representation of the Petersen graph, about whether it's illustrative enough to even be worth it, about life in general...whatever.