In our last project we played around with a really terrific site shared by Bill Hanage which shows how a virus can spread across a network:

That project is here:

https://mikesmathpage.wordpress.com/2020/09/10/having-my-younger-son-explore-fantastic-website-shared-by-bill-hanage-explaining-how-social-distancing-changes-network-connections/

Following that project, I thought my son would enjoy seeing different types of graphs (much smaller ones) and the different ways those graphs can be represented. I showed him some simple commands in Mathematica that would allow him to play around with these simple graphs and asked him to show 4 that he found interesting.

He started with a plain vanilla triangular graph. I was a little surprised that he wanted to start with a basic example like this one, but it ended up leading to a really nice discussion:

The next graph he thought was interesting is called the Levi Graph. I haven’t looked to see where this graph comes from, but the different ways of representing it were fascinating.

The next graph he picked is called the Gem Graph. This one is easier to understand than the Levi Graph as it has only 2 different representations. We had a good discussion about how to see that those two representations were the same:

Finally, for the last example, he chose the Icosahedral Graph. This is another graph with many different representations – some of which are really cool! It is hard to believe that all of these graphs are the same, and that fact / surprise led to a fun discussion:

Definitely a fun project. It is fun to see how kids react to seeing graphs / networks. This year I think we’ve all learned one critical application of the ideas in graph theory is understanding how a virus spreads, so I think helping kids see some ideas in graph theory is important. Playing around with different types of graphs definitely makes for a fun introduction to the subject.