Regularity, depth and arithmetic rank of bipartite edge ideals

By Manoj Kummini. 2009

99 citations recorded by the source corpus.

We study minimal free resolutions of edge ideals of bipartite graphs. We associate a directed graph to a bipartite graph whose edge ideal is unmixed, and give expressions for the regularity and the depth of the edge ideal in terms of invariants of the directed graph. For some classes of unmixed edge ideals, we show that the arithmetic rank of the ideal equals projective dimension of its quotient.

Subfields: Algebra and Number Theory, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/9e7909dd2d72e5e266ce09ba539a1a8480e97ea3

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Algebra and Number Theory

In Geometry and Topology

Citations in the corpus

Edge ideals: algebraic and combinatorial properties (2010, background): “ns (i) and (ii) of Theorem 3.14 are satisﬁed. Corollary 3.15 shows that the regularity of R/I(G) equals im(G) for any forest G, which was ﬁrst proved by Zheng [118]. If G is an unmixed graph, Kummini [77] showed that reg(R/I(G)) equals the induced matching number of G. If G is claw-free and its comp”
Matchings, coverings, and Castelnuovo-Mumford regularity (2010, background): “(Kumini [24]) If G is a well-covered bipartite graph, then reg (R/I(G)) = indmatchG.”
Algebraic study on Cameron-Walker graphs (2013, background): “([11, Lemma 2.2] and [9, Theorem 6.7]) that im(G) reg(S=I(G)) m(G): One has reg(S=I(G)) = im(G) for, e.g., chordal graphs, unmixed bipartite graphs and sequentially Cohen{Macaulay bipartite graphs ([9, 13, 17]; see also [7, 15, 21, 22]). Cameron and Walker ([3, Theorem 1]) gave a classication of the”
Vertex decomposability and regularity of very well-covered graphs (2010, background): “These results improve the results of [2], [9], and [11].”
Regularity of Edge Ideals and Their Powers (2017, background): “following cases: 15 (a) Gis a chordal graph (see [29]); (b) Gis a weakly chordal graph (see [64]); (c) Gis sequentially Cohen-Macaulay bipartite graph (see [62]); (d) Gis unmixed bipartite graph (see [43]); (e) Gis very well-covered graph (see [46]); (f) Gis vertex decomposable graph and has no clos”
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): “In particular, there has been active research on bounding the homological invariants of edge ideals in terms of the combinatorial invariants of the associated graphs, see for example [3,10,12,15,17,19,22,23,26,28,32,33].”
The regularity of powers of edge ideals (2014, background): “hen is it true that for all s≥ 2, I(G)s has linear minimal free resolution? Bounds on the regularity of edge ideals have been studied by a number of researchers (see [1], [2], [3], [4], [5], [6], [7],[8] [9], [11],[12],[13]). For example, Fr¨oberg (see [3]) has shown that, when I(G) is the edge idea”
Regularity of Squarefree Monomial Ideals (2013, background): “ the maximum size of an induced matching in G. Then reg(G) = (G) + 1 in the following cases: (1) Gis a sequentially Cohen-Macaulay bipartite graph (see [48]); (2) Gis an unmixed bipartite graph (see [36]); (3) Gis a very well-covered graph (see [38]); (4) Gis a C5-free vertex decomposable graph (see”
Bounds on the regularity and projective dimension of ideals associated to graphs (2011, background): “It should be noted that the projective dimension and regularity of Stanley-Reisner ideals, or more specifically edge ideals, have attracted a lot of attention; see [3, 10, 11, 12, 16, 21, 23] and references given therein.”
Depths of Powers of the Edge Ideal of a Tree (2009, background): “Then depth R/I has been studied by various authors (see for example [8, 9, 15]).”