A new graph invariant arises in toric topology

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