A new graph invariant arises in toric topology: rational Betti numbers of real toric varieties arising from graphs

We introduce new combinatorial invariants of any finite simple graph, which arise in toric topology. We compute the i-th rational Betti number and Euler characteristic of the real toric variety associated to a graph associahedron. They can be calculated by a purely combinatorial method in terms of graphs. To our surprise, for specific families of the graph, our invariants are deeply related to well-known combinatorial sequences such as the Catalan numbers and Euler zigzag numbers.