Betti Numbers of Graph Ideals

By Sean Jacques. 2004

159 citations recorded by the source corpus.

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.

Subfields: Algebra and Number Theory, Computational Theory and Mathematics

View Paper: https://www.semanticscholar.org/paper/37a2705319a530d9942076797ffb471bc1504c62

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In Algebra and Number Theory

In Computational Theory and Mathematics

Citations in the corpus

A CLASS OF HYPERGRAPHS THAT GENERALIZES CHORDAL GRAPHS (2008, background): “the minimal non faces of a simplicial complex or, equally well, the relations in the k-algebra R/I(H). The connections between a (hyper)graph and its independence complex are explored in, for example [4, 7, 13]. Another way to use hypergraphs to investigate the properties of simplicial complexes was”
Regularity of Edge Ideals and Their Powers (2017, methodology): “= bn 3 cbe the induced matching number of C n. Then regI= ˆ + 1 if n0;1 (mod 3) + 2 if n2 (mod 3) and for any q2, we have regIq= 2q+ 1: Sketch of proof. The rst statement was already proved in [36]. To prove the second statement, again thanks to the general lower bound of Theorem 5.1, it remains to”
Resolutions of square-free monomial ideals via facet ideals: a survey (2006, methodology): “For instance, Jacques [31] provided formulas for the graded Betti numbers of edge ideals of special classes of graphs including cycles and complete graphs; Roth and the second author [37] calculated the graded Betti numbers in the linear strand of edge ideals of graphs having no minimal cycle of len”
Splittable ideals and the resolutions of monomial ideals (2005, background): “An examination of the papers [3, 11, 17, 18, 19, 21, 27] reveals that these formulas provide the basis for most of the known results on the numbers βi,j(I(G)).”
Monomial ideals, edge ideals of hypergraphs, and their graded Betti numbers (2006, background): “Restricted to simple graphs, we obtain the following interesting corollary, which extends a result of [19, 20] (that proved the result for forests).”
Non-vanishingness of Betti numbers of edge ideals (2011, background): “ Also, the (Castelnuovo–Mumford) regularity of S/I(G) is deﬁned by regS/I(G) := max{j − i : βi,j(S/I(G)) 6= 0 }. In this paper, we focus on these invariants, which have studied by many authors, e.g., [2, 7, 9, 10, 11, 12, 14, 16]. In particular, Zheng [16] characterized the projective dimension and ”
Regularity of powers of bipartite graphs (2016, unknown): “It was shown by Jacques, [17], that if n ≡ 2(mod 3), then reg(I (Cn)) = ν(Cn)+ 2, and Beyarslan, Hà and Trung proved that reg(I (Cn)s) = 2s + ν(Cn) − 1 for all s ≥ 2 [5].”
Algebraic Properties of the Path Ideal of a Tree (2009, background): “Our subclass contains the edge ideals of trees, and thus, our formula generalizes a result of Jacques and Katzman [12].”