The facet ideals of matching complexes of line graphs (original) (raw)
Projective dimension and regularity of the path ideal of the line graph
arXiv (Cornell University), 2016
By generalizing the notion of the path ideal of a graph, we study some algebraic properties of some path ideals associated to a line graph. We show that the quotient ring of these ideals are always sequentially Cohen-Macaulay and also provide some exact formulas for the projective dimension and the regularity of these ideals. As some consequences, we give some exact formulas for the depth of these ideals.
Projective dimension and regularity of edge ideals of some weighted oriented graphs
Rocky Mountain Journal of Mathematics, 2019
In this paper we provide some exact formulas for the projective dimension and the regularity of edge ideals associated to three special types of vertex-weighted oriented m-partite graphs. These formulas are functions of the weight and number of vertices. We also give some examples to show that these formulas are related to direction selection and the weight of vertices.
The square of the line graph and path ideals
Mathematical Communications, 2017
For path ideals of the square of the line graph we compute the Krull dimension, we characterize the linear resolution property in combinatorial terms. We bound the Castelnuovo-Mumford regularity and the projective dimension in terms of the corresponding invariants of two sub-hypergraph. We present some open questions.
Regularity and projective dimension of the edge ideal of C_5C_5C_5-free vertex decomposable graphs
Proceedings of the American Mathematical Society, 2014
In this paper, we explain the regularity, projective dimension and depth of the edge ideal of some classes of graphs in terms of invariants of graphs. We show that for a C 5-free vertex decomposable graph G, reg(R/I(G)) = c G , where c G is the maximum number of 3-disjoint edges in G. Moreover, for this class of graphs we characterize pd(R/I(G)) and depth(R/I(G)). As a corollary we describe these invariants in forests and sequentially Cohen-Macaulay bipartite graphs.
On the Facet Ideal of an Expanded Simplicial Complex
Bulletin of the Iranian Mathematical Society, 2018
For a simplicial complex ∆, the affect of the expansion functor on combinatorial properties of ∆ and algebraic properties of its Stanley-Reisner ring has been studied in some previous papers. In this paper, we consider the facet ideal I(∆) and its Alexander dual which we denote by J∆ to see how the expansion functor alter the algebraic properties of these ideals. It is shown that for any expansion ∆ α the ideals J∆ and J∆α have the same total Betti numbers and their Cohen-Macaulayness are equivalent, which implies that the regularities of the ideals I(∆) and I(∆ α) are equal. Moreover, the projective dimensions of I(∆) and I(∆ α) are compared. In the sequel for a graph G, some properties that are equivalent in G and its expansions are presented and for a Cohen-Macaulay (resp. sequentially Cohen-Macaulay and shellable) graph G, we give some conditions for adding or removing a vertex from G, so that the remaining graph is still Cohen-Macaulay (resp. sequentially Cohen-Macaulay and shellable).
Bounds on the regularity and projective dimension of ideals associated to graphs
Journal of Algebraic Combinatorics, 2012
In this paper we give new upper bounds on the regularity of edge ideals whose resolutions are k-steps linear; surprisingly, the bounds are logarithmic in the number of variables. We also give various bounds for the projective dimension of such ideals, generalizing other recent results. By Alexander duality, our results also apply to unmixed square-free monomial ideals of codimension two. We also discuss and connect these results to more classical topics in commutative algebra.
The facet ideals of chessboard complexes
arXiv (Cornell University), 2022
In this paper we describe the irreducible decomposition of the facet ideal F (∆ m,n ) of the chessboard complex ∆ m,n with n ≥ m. We also provide some lower bounds for depth and regularity of the facet ideal F (∆ m,n ). When m ≤ 3, we prove that these lower bounds can be obtained.
Line graphs of simplicial complexes
2022
We consider the line graph of a simplicial complex. We prove that, as in the case of line graphs of simple graphs, one can compute the second graded Betti number of the facet ideal of a pure simplicial complex in terms of the combinatorial structure of its line graph. We also give a characterization of those graphs which are line graphs of some pure simplicial complex. In the end, we consider pure simplicial complexes which are chordal and study their relation to chordal line graphs.
Journal of Algebra and Its Applications, 2019
In this paper, we provide some precise formulas for regularity of powers of edge ideal of the disjoint union of some weighted oriented gap-free bipartite graphs. For the projective dimension of such an edge ideal, we give its exact formula. Meanwhile, we also give the upper and lower bounds of projective dimension of higher powers of such an edge ideal. As an application, we present regularity and projective dimension of powers of edge ideal of some gap-free bipartite undirected graphs. Some examples show that these formulas are related to direction selection.
The line graph of a tree and its edge ideal
2021
We describe all the trees with the property that the corresponding edge ideal of their line graph has a linear resolution. As a consequence, we give a complete characterization of those trees T for which the line graph L(T ) is cochordal. We compute also the second Betti number of the edge ideal of L(T ) and we determine the number of cycles in L(T ). As a consequence, we obtain also the first Zagreb index of a graph. For edge ideals of line graphs of caterpillar graphs we determine the Krull dimension, the Castelnuovo-Mumford regularity, and the projective dimension under some additional assumption on the degrees of the cutpoints.