Betti Numbers of Graph Ideals

TL;DR

This paper explores the algebraic properties of graph-related monomial ideals, specifically Betti numbers and projective dimensions, providing explicit formulas and combinatorial interpretations for cycles and forests.

Contribution

It introduces new combinatorial invariants for forests based on algebraic Betti numbers and offers explicit descriptions for cycles and forests.

Findings

Explicit Betti number formulas for cycles and forests

A combinatorial method to determine projective dimension of forests

New numerical invariants for graphs derived from algebraic properties

Abstract

In this thesis we investigate certain types of monomial ideals of polynomial rings over fields. We are interested in minimal free resolutions of these ideals (or equivalently the quotients of the polynomial ring by the ideals) considered as modules over the polynomial ring. There is no simple method of finding such resolutions but in the case of Stanley-Reisner ideals Hochster's formula and its variants provide a way to compute the Betti numbers of these resolutions. Even with these formulae it is not in general possible to find especially explicit or useful descriptions of the Betti numbers. However we restrict our attention to those ideals which are generated by square free monomials of degree 2. The purpose of this is to associate these ideals with graphs. This provides a link between algebraic objects, the monomial ideals, and combinatorial objects, the graphs. This correspondence enables us do define new numerical invariants of graphs: the Betti numbers and projective dimension of the corresponding graph ideals. We find explicit descriptions of the Betti numbers and projective dimensions of cycles and forests. In the case of forests we find a method of describing the Betti numbers in terms of the Betti numbers of subforests. This also leads to a description of the projective dimension of a forests in terms of the projective dimensions of its subforests. It turns out that the projective dimension of forests can be defined in purely combinatorial terms and hence it gives a new combinatorial numerical invariant of forests.