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 and Euler characteristic of the real toric variety associated to a graph associahedron P B(G) . They can be calculated by a purely combinatorial method (in terms of graphs) and are named ai(G) and b(G), respectively. 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.

Small covers of graph-associahedra and realization of cycles

“…There is an unpublished formula by Suciu and Trevisan that reduces this computation to the computation of the homology groups for certain unions of faces of P . For small covers of graph-associahedra M P Γ ,λcan corresponding to the canonical characteristic function λ can coming from the Delzant structure, Choi and Park [11] obtained an explicit formula for the Betti numbers with coefficients in Q from special invariants of the graph Γ. However, even in this case the problem of finding a graph with the smallest total Betti number of M P Γ ,λcan remains unsolved, though it is very likely that the minimum is attained for the path graph L n+1 .…”

“…For n ≥ 2, the manifold M Pe n ,λcan is non-orientable. Its Betti numbers with coefficients in Q were computed by Henderson [26] (see also [11]):…”

Toric Fano varieties associated to building sets

“…It is natural to ask how geometric properties of the toric variety associated to a graph associahedron or a graph cubeahedron translate into properties of the graph. The rational Betti numbers of the real toric manifold, the set of real points in the associated toric variety, are computed in [3] for a graph associahedron, and in [8] for a graph cubeahedron. A nonsingular projective variety is said to be Fano (resp.…”

Shellable posets arising from the even subgraphs of a graph

“…For a simple graph G, it was shown in [8] that for each simplicial subcomplex K S of K X G , there is a subgraph H of G such that K S is homotopy equivalent to the order complex of the proper part of the poset of even subgraphs of H. The work of [8] was generalized to a graph (allowing multiple edges) in [10]. A graph H is a partial underlying induced graph (PI-graph for short) of a graph G if H can be obtained from an induced subgraph of G by replacing some bundles with simple edges, where a bundle is a maximal set of multiple edges which have the same pair of endpoints.…”

“…Remark. In [8], Theorem 3.2 is used to determine the homotopy type of the order complex ∆(P even H ). Finally, ∆(P even H ) is homotopy equivalent to a wedge of the same dimensional spheres, and the Möbius invariant 5 µ(P even H ) is equal to the (ℓ − 2)th Betti number of ∆(P even H ), where ℓ is the length of the poset P even H .…”