Graph invariants and betti numbers of real toric manifolds - Park, Boram; Park, Hanchul; Park, Seonjeong (Osaka Journal of Mathematics, 2020)
For a graph G, the graph cubeahedron □G and the graph associahedron △G are simple convex polytopes which admit (real) toric manifolds. In this paper, we introduce a graph invariant, called the b-number, and show that the b-numbers compute the Betti numbers of the real toric manifold XR(□G ) corresponding to □G . The b-number is a counterpart of the notion of anumber, introduced by S. Choi and the second named author, which computes the Betti numbers of the real toric manifold XR(△G ) corresponding to △G . We also study various relationships between a-numbers and b-numbers from the viewpoint of toric topology. Interestingly, for a forest G and its line graph L(G), the real toric manifolds XR(△G )andXR(□L(G) )havethesame Betti numbers.