The $f$--vector of the clique complex of chordal graphs and Betti numbers of edge ideals of uniform hypergraphs

TL;DR

This paper investigates algebraic and combinatorial properties of chordal graphs and uniform hypergraphs, providing formulas and inequalities for Betti numbers, the $f$-vector, and multiplicity of associated algebraic structures.

Contribution

It introduces explicit formulas and inequalities for Betti numbers, $f$-vectors, and multiplicities related to chordal graphs and uniform hypergraphs, advancing understanding of their algebraic invariants.

Findings

Betti numbers of edge ideals with linear resolutions are characterized.

Inequalities and algebraic equations for the $f$-vector of clique complexes are established.

Explicit formulas for the multiplicity of Stanley-Reisner rings of chordal graphs are provided.

Abstract

We describe the Betti numbers of the edge ideals $I(G)$ of uniform hypergraphs $G$ such that $I(G)$ has linear graded free resolution. We give an algebraic equation system and some inequalities for the components of the $f$--vector of the clique complex of an arbitrary chordal graph. Finally we present an explicit formula for the multiplicity of the Stanley-Reisner ring of the edge ideals of any chordal graph.