Tutte-Coxeter Graph

I don't know any examples, and maybe there aren't (at least as of today). But it is beautiful, deep and fun at the same time, so if you're into that sort of thing it's a perfect hobby.

Some may argue it do improve someones life as pure mathematical elegance, even if not 'applied' to the 'real' world.

Automorphisms of strange structures like this are connected with the constellation diagrams that got us from 4.8 K bits/sec up to 57.6 K bits/sec and beyond on modems. They're also used in designing block codes and error correction codes.

The interactions between graph theory and group theory are deep and productive, but not immediately obvious, and occasionally convoluted. There are others here who can talk more knowledgeably about the details - not my area.

It's also the case that pursuing these sorts of things purely for the sake of solving the puzzle can turn out to be of value, even if there are no immediate and obvious applications. Graph Theory and Combinatorics were just curiosities originally, but have turned out to be critical in many, many areas, not least coding theory, processor design, algorithm analysis, and much, much more.

I wrote about this some time ago, and have submitted a link so people can comment and/or write more. It's here:

https://news.ycombinator.com/item?id=10066365

It will probably sink without trace, but thought you might be interested.

I guess at the very least it can serve as a useful 'edge case' test for an implementation of an algorithm operating on a graph. E.g. the Peterson graph[1] is also quite useful for that.

[1] https://en.wikipedia.org/wiki/Petersen_graph