M Theory Lesson 163

I was delighted to come across a remarkable paper by Bolker, Guillemin and Holm, entitled How is a graph like a manifold? The paper begins with some remarks on Stanley's proof of McMullen's conjecture and then moves onto geometric definitions and problems involving the Betti numbers of graphs.

For example, the graph of the symmetric group $S_{n}$, that is the permutohedron, corresponds to the flag manifold of subspaces of $\mathbb{CP}^{n}$. In M Theory we are particularly interested in $S_{3}$, a hexagon including the diagonals, which has three geodesics, one of which is shown in the diagram.