2/12/21

First, 2/12/21, 21221, 21 [2] 21.

Kind of a nice date, not as good as some but not bad. 

Now, onto the musings.

I've begun playing around with graphs, nothing much just playing. Hopefully I'll be doing some research in graph theory this summer so I've just started to learn a little bit here and there. Mainly, I'm trying to familiarize myself with the pictures. Getting over the notation as quickly as possible and finding new patterns wherever I can.

So, what have I done?

Well, since I don't know anything about graph theory, I first started out with trying to translate something else I know into graphs. I chose the platonic solids because I think they're neat. I began with a tetrahedron, or a D4 as I think of it. 

The tetrahedron has four points and six edges, which I translated to four nodes and six...edges. It was starting with the tetrahedron that I thought of a really interesting metric to judge the graphs by. That metric was minimum and maximum closed path. I defined closed path to mean a path from one node, along unique edges (no repeating) and back to the original node. It was also with the tetrahedron that I began to think about distance between neighboring nodes.

For the tetrahedron, each node is connected to three other nodes. I originally thought this would be true for all platonic solids but it was later proven false. However, that was one of my first conjectures.

Conjecture 1: Each node of a platonic solid is connected to three unique nodes.

I also noticed how each node's neighbor was connected to the firs node. Like the center node, going up from the center and then out to either side takes you to another neighbor of the center node. The tetrahedron is a really tight nit cluster of nodes and edges.

Each node has a distance of zero from the other nodes (they're connected by an edge).

The minimum closed path for the tetrahedron is three. The maximum closed path I could find was four. 

The second shape I looked at was the cube. This was just because it was the second simplest platonic solid for me to work with. I played around with drawing different kinds of graphs for the cube before settling on this one. I like it a lot since it reflects how we draw cubes normally with perspective and it was reminiscent of 2D drawings of 4D cubes I drew over the summer.

For the cube, each node is again connected to three other nodes. In 3D space, the cube will jut out in each available axis (like with a square, 2D cube, a cube, 3D cube, and a hypercube, +4D cube). On a graph though, a 3D cube is just represented by have eight nodes, 12 edges, having each node connected to three unique nodes, and the extra condition that none of its neighbors are neighbors with each other. That became apparent when I tried drawing the graph of a cube in the abstract. Weird stuff.

Each node has a distance 2 away from every other node, except for that nodes nemesis who is 3 away.

The minimum closed path for a cube is 4, which is nice when you consider the square. The maximum closed path I could find was 8.

The final graph I played with this week was octahedron, or D8. The D8 is composed of two pyramids stuck square end to square end. It differs from the tetrahedron in the fact that it has a square base and not a triangular one. In graph representation, I decided on a hexagonal description with intersecting edges. I don't know if intersecting edges are important to graph theory but since I could have just as easily drawn them going around the outside, I decided it was trivial. 

The D8 was the weirdest shape by far which I think is because its not just made out of squares or triangles. Looking at the graph above you can identify several triangles and several squares. This one makes me want to spend more time trying to figure out how to multiply "2D" graphs to see if I can make some more "3D" graphs. The D8 has 6 nodes and 12 edges. It's weird because it has two nodes that have four neighbors, disproving my first conjecture. The constraints on this one are really weird too. The only ones I could think of that truly define the graph uniquely are, minimum path of 3 and maximum path of 7. I don't know if that truly holds but I'll conjecture it.

Conjecture 2: Any graph with a minimum path of 3 and a maximum path of 7 from any node on the graph is isomorphic to an octahedron.

I'll explore these things more in the future.

The final thing I explored with my platonic graphs was multiplication. This was based purely off of a single numberphile video I watched on the topic.

https://www.youtube.com/watch?v=Tnu_Ws7Llo4&ab_channel=Numberphile

Great video.

From my understanding, when multiplying two graphs you replace the nodes of one graph with the entirety of the other graph. I tried to do an example with my tetrahedrons.

In the picture, you can see the read nodes being filled by the blue graphs. The number of nodes is 16 and the number of edges is now 30. I haven't done enough examples to find a pattern but it'll be pretty obvious I'm sure. Something like #nodes * (#edges + 1) or something. Idk.

Goals for future exploration:

1. Go back and examine the relationship between multiplying "2D" graphs to see if you can get "3D" graphs.

2. Find a relationship between platonic solids, nodes, and edges.

3. Find special graph specific qualities of these platonic graphs.

4. Explore the relationship of multiplying graphs in general. Find some basic principle.

That's all for this weeks musings.

Ciao!