Betti numbers of chordal graphs and $f$-vectors of simplicial complexes

Details Betti numbers of chordal graphs and $f$-vectors of simplicial complexes Betti numbers of chordal graphs and $f$-vectors of simplicial complexes

2009-07-28

arxiv.org/abs/0907.4839v1

Mathematics

Combinatorics

15 pages

Scientific paper

Let $G$ be a chordal graph and $I(G)$ its edge ideal. Let $\beta (I(G)) = (\beta_0, \beta_1, ..., \beta_p)$ denote the Betti sequence of $I(G)$, where $\beta_i$ stands for the $i$th total Betti number of $I(G)$ and where $p$ is the projective dimension of $I(G)$. It will be shown that there exists a simplicial complex $\Delta$ of dimension $p$ whose $f$-vector $f (\Delta) = (f_0, f_1, ..., f_p)$ coincides with $\beta (I(G))$.

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