## Vertex.

I mentioned graph theory several weeks back, when this Mathematics A To Z project was barely begun. It’s a fun field. It’s a great one for doodlers, and it’s one that has surprising links to other problems.

Graph theory divides the conceptual universe into “things that could be connected” and “ways they are connected”. The “things that could be connected” we call vertices. The “ways they are connected” are the edges. Vertices might have an obvious physical interpretation. They might, represent the corners of a cube or a pyramid or some other common shape. That, I imagine, is why these things were ever called vertices. A diagram of a graph can look a lot like a drawing of a solid object. It doesn’t have to, though. Many graphs will have vertices and edges connected in ways that no solid object could have. They will usually be ones that you could build in wireframe. Use gumdrops for the vertices and strands of wire or plastic or pencils for the edges.

Vertices might stand in for the houses that need to be connected to sources of water and electricity and Internet. They might be the way we represent devices connected on the Internet. They might represent all the area within a state’s boundaries. The Köningsburg bridge problem, held up as the ancestor of graph theory, has its vertices represent the islands and river banks one gets to by bridges. Vertices are, as I say, the things that might be connected.

“Things that might be connected” is a broader category than you might imagine. For example, an important practical use of mathematics is making error-detecting and error-correcting codes. This is how you might send a message that gets garbled — in sending, in transmitting, or in reception — and still understand what was meant. You can model error-detecting or correcting codes as a graph. In this case every possible message is a vertex. Edges connect together the messages that could plausibly be misinterpreted as one another. How many edges you draw — how much misunderstanding you allow for — depends on how many errors you want to be able to detect, or to correct.

When we draw this on paper or a chalkboard or the like we usually draw it as a + or an x or maybe a *. How much we draw depends on how afraid we are of losing sight of it as we keep working. In publication it’s often drawn as a simple dot. This is because printers are able to draw dots that don’t get muddied up by edges being drawn in or eraser marks removing edges.

When an uninitiated undergraduate student asks me what graph theory is all about, my favorite illustration is the “Six Degrees of Kevin Bacon” problem. I ask them visualize each actor/actress as a point (vertex), and then connect a line between two vertices if the two actors/actresses appeared in the same movie. While this is a fairly abstract graph, most students (in my experience) are able to visualize the graph and understand the Kevin Bacon problem as finding the least “distance” from a vertex to the Kevin Bacon vertex.

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Oh, thank you. I hadn’t thought of that example but you’re right, it works very well.

I’m actually a bit surprised it does work so well, since the graph of actors-sharing-movies is such a complicated and entangled one. But then people come to it having used the idea a lot, so it’s nicely familiar.

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