## Preorders and Finite Topological Spaces

Today I tweeted that I had asked my topology students to find all of the different topologies of a two-point set and a three-point set. It turns out that there are three of the former and nine of the latter. (The sequence of the number of topologies for an n-point set begins 1 (for ),…

## What’s in a name?

I had an interesting interaction on Twitter today. I’m doing some research with a student that involves a little graph theory. We needed some graph theory terminology that I was sure was well-known (but not known to us), so I turned to Twitter. Let be a vertex of directed graph. Define to be the set…

## An application of graph theory to architecture

Several years ago I came across a fascinating application of graph theory to architecture. It is in the 1983 book Incidence and symmetry in design and architecture, by Jenny A. Baglivo and Jack E. Graver. I don’t know if it is well known among experts in the field, but I’ve never seen it elsewhere. So…

## Graph theory is funny

I read and laugh at Jessica Hagy’s Indexed every day. Today Jessica created the following index card titled 7 wonders of the modern world which is based on , the complete graph on 7 vertices. She has created similar index cards before, such as… : For my MBA friends (apologies to Mr. Porter) : Yes,…

## Königsberg today

In 1736 Leonhard Euler solved the now-famous bridges of Königsberg problem.  It is often hailed as the birth of topology and graph theory. Although no graph appears in Euler’s paper, his argument is topological in spirit. (He proved another famous topological theorem a decade and a half later.) The problem is stated as follows. If a resident of Königsberg…