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.
Acknowledgements. The authors sincerely thank the referee for her/his valuable comments which gave improvements of this paper. The authors also thank Prof. Vincent Pilaud and Dr. Thibault Manneville for their kind answer to the questions for the compatibility fans. Finally, we thank Yusuke Suyama for his valuable comment. This work was partially supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Korean Government (NRF-2018R1C1B6003577 to B. P.), (NRF-2019R1G1A1007862 to H. P.), and (NRF-2018R1A6A3A11047606 to S. P. (corresponding author)).
