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

TL;DR

This paper establishes a connection between the Betti numbers of edge ideals of chordal graphs and the $f$-vectors of certain simplicial complexes, revealing a combinatorial-algebraic correspondence.

Contribution

It proves that for any chordal graph, its edge ideal's Betti sequence matches the $f$-vector of a specific simplicial complex, linking algebraic invariants to combinatorial structures.

Findings

Existence of a simplicial complex with an $f$-vector equal to the Betti sequence of the edge ideal.

Betti numbers of chordal graph edge ideals can be realized as $f$-vectors of simplicial complexes.

Provides a combinatorial interpretation of algebraic invariants of chordal graphs.

Abstract

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 $\beta (I(G))$.