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Choi, Suyoung; Park, Hanchul

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

J. Math. Soc. Japan 67, No. 2, 699-720 (2015).
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.
Reviewer: Michael Wiemeler (Augsburg)

Cited in 3 Reviews
Cited in 12 Documents

MSC:

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

Keywords:

graph associahedron; real toric variety; graph invariant; poset topology; shellable poset

Software:

OEIS
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\textit{S. Choi} and \textit{H. Park}, J. Math. Soc. Japan 67, No. 2, 699--720 (2015; Zbl 1326.57044)
Full Text: DOI arXiv Euclid

Online Encyclopedia of Integer Sequences:

Irregular triangle read by rows: one half of the coefficients of the expansion of (2*x)^n in terms of Chebyshev T-polynomials.
Catalan triangle read by rows. Also triangle of expansions of powers of x in terms of Chebyshev polynomials U_n(x).
Nonzero coefficients of the Swiss-Knife polynomials for the computation of Euler, tangent, and Bernoulli numbers (triangle read by rows).
Triangle read by rows: k-th ”a-number” of star graph K_{1,n-1}.

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