A new graph invariant arises in toric topology

Abstract

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.