A new graph invariant arises in toric topology. (English) Zbl 1326.57044

For a finite simple graph \(G\), the graph associahedron \(P_{\mathcal{B}(G)}\) is a Delzant polytope. To this polytope one can associate a real toric variety \(M(G)\). In the paper under review a purely combinatorial formula for the rational Betti numbers of \(M(G)\) is given.
The authors define new combinatorial graph invariants which they call \(a\)-numbers and show that the \(i\)-th \(a\)-number of \(G\) is equal to the \(i\)-th Betti number of \(M(G)\). The proof of this result is based on a formula for the rational Betti numbers of a real toric manifold due to A. Suciu and A. Trevisan [“Real toric varieties and abelian covers of generalized Davis–Januszkiewicz spaces”, Preprint, (2012)] and an investigation of the combinatorics of \(P_{\mathcal{B}(G)}\).
Moreover, the \(a\)-numbers of graphs \(G\) in special families are computed. For example, it is shown that the \(a\)-numbers of a complete graph or a star graph are related to the Euler zigzag numbers. Furthermore, the \(a\)-numbers of a path graph are related to the Catalan-numbers.


57N65 Algebraic topology of manifolds
55U10 Simplicial sets and complexes in algebraic topology
05C30 Enumeration in graph theory


Full Text: DOI arXiv Euclid


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