Binomial Edge Ideals Research Articles

Binomial edge ideals of bipartite graphs

Binomial edge ideals are a noteworthy class of binomial ideals that can be associated with graphs, generalizing the ideals of 2-minors. For bipartite graphs we prove the converse of Hartshorne’s Connectedness Theorem, according to which if an ideal is Cohen–Macaulay, then its dual graph is connected. This allows us to classify Cohen–Macaulay binomial edge ideals of bipartite graphs, giving an explicit and recursive construction in graph-theoretical terms. This result represents a binomial analogue of the celebrated characterization of (monomial) edge ideals of bipartite graphs due to Herzog and Hibi (2005).

On the hilbert series of binomial edge ideals of generalized trees

In this paper we introduce the concept of generalized trees and compute the Hilbert series of their binomial edge ideals‎.

On the binomial edge ideals of block graphs

Abstract We find a class of block graphs whose binomial edge ideals have minimal regularity. As a consequence, we characterize the trees whose binomial edge ideals have minimal regularity. Also, we show that the binomial edge ideal of a block graph has the same depth as its initial ideal.

On closed graphs, II

A graph is closed when its vertices have a labeling by [n] with a certain property first discovered in the study of binomial edge ideals. In this article, we explore various aspects of closed graphs, including the number of closed labelings and clustering coefficients.

Parity binomial edge ideals

Parity binomial edge ideals of simple undirected graphs are introduced. Unlike binomial edge ideals, they do not have square-free Gr\"obner bases and are radical if only if the graph is bipartite or the characteristic of the ground field is not two. The minimal primes are determined and shown to encode combinatorics of even and odd walks in the graph. A mesoprimary decomposition is determined and shown to be a primary decomposition in characteristic two.

The Castelnuovo–Mumford regularity of binomial edge ideals

We prove a conjectured upper bound for the Castelnuovo–Mumford regularity of binomial edge ideals of graphs, due to Matsuda and Murai. Indeed, we prove that reg(JG)≤n−1 for any graph G with n vertices, which is not a path.

Linear flags and Koszul filtrations

We show that the graded maximal ideal of a graded $K$-algebra $R$ has linear quotients for a suitable choice and order of its generators if the defining ideal of $R$ has a quadratic Gr\"obner basis with respect to the reverse lexicographic order, and show that this linear quotient property for algebras defined by binomial edge ideals characterizes closed graphs. Furthermore, for algebras defined by binomial edge ideals attached to a closed graph and for join-meet rings attached to a finite distributive lattice we present explicit Koszul filtrations.

Some Cohen–Macaulay and Unmixed Binomial Edge Ideals

We study unmixed and Cohen-Macaulay properties of the binomial edge ideal of some classes of graphs. We compute the depth of the binomial edge ideal of a generalized block graph. We also characterize all generalized block graphs whose binomial edge ideals are Cohen–Macaulay and unmixed. So that we generalize the results of Ene, Herzog, and Hibi on block graphs. Moreover, we study unmixedness and Cohen–Macaulayness of the binomial edge ideal of some graph products such as the join and corona of two graphs with respect to the original graphs.

Binomial Edge Ideals with Special Set of Associated Primes

In this paper, we show when the intersection of two binomial edge ideals is again a binomial edge ideal. Using it, we study binomial edge ideal J G under the assumption that either |Ass(J G )| ≤3 or Ass(J G )∖{J K n } is a set of monomial ideals.

On the regularity of binomial edge ideals

We study the regularity of binomial edge ideals. For a closed graph G we show that the regularity of the binomial edge ideal JG coincides with the regularity of inlex(JG) and can be expressed in terms of the combinatorial data of G. In addition, we give positive answers to Matsuda-Murai conjecture (8) for some classes of graphs.

Binomial edge ideals with pure resolutions

We characterize all graphs whose binomial edge ideals have pure resolutions. Moreover, we introduce a new switching of graphs which does not change some algebraic invariants of graphs, and using this, we study the linear strand of the binomial edge ideals for some classes of graphs. Also, we pose a conjecture on the linear strand of such ideals for every graph.

The binomial edge ideal of a pair of graphs

AbstractWe introduce a class of ideals generated by a set of 2-minors of an (m×n)-matrix of indeterminates indexed by a pair of graphs. This class of ideals is a natural common generalization of binomial edge ideals and ideals generated by adjacent minors. We determine the minimal prime ideals of such ideals and give a lower bound for their degree of nilpotency. In some special cases we compute their Gröbner basis and characterize unmixedness and Cohen–Macaulayness.

The binomial edge ideal of a pair of graphs

AbstractWe introduce a class of ideals generated by a set of 2-minors of an (m×n)-matrix of indeterminates indexed by a pair of graphs. This class of ideals is a natural common generalization of binomial edge ideals and ideals generated by adjacent minors. We determine the minimal prime ideals of such ideals and give a lower bound for their degree of nilpotency. In some special cases we compute their Gröbner basis and characterize unmixedness and Cohen–Macaulayness.

Construction of Cohen–Macaulay Binomial Edge Ideals

We discuss algebraic and homological properties of binomial edge ideals associated with the graphs which are obtained by gluing of subgraphs and the formation of cones.

Generalized binomial edge ideals

This paper studies a class of binomial ideals associated to graphs with finite vertex sets. They generalize the binomial edge ideals, and they arise in the study of conditional independence ideals. A Gröbner basis can be computed by studying paths in the graph. Since these Gröbner bases are square-free, generalized binomial edge ideals are radical. To find the primary decomposition a combinatorial problem involving the connected components of subgraphs has to be solved. The irreducible components of the solution variety are all rational.

Binomial Edge Ideals of Graphs

We characterize all graphs whose binomial edge ideals have a linear resolution. Indeed, we show that complete graphs are the only graphs with this property. We also compute some graded components of the first Betti number of the binomial edge ideal of a graph with respect to the graphical terms. Finally, we give an upper bound for the Castelnuovo-Mumford regularity of the binomial edge ideal of a closed graph.

Cohen-Macaulay binomial edge ideals

AbstractWe study the depth of classes of binomial edge ideals and classify all closed graphs whose binomial edge ideal is Cohen-Macaulay.

Cohen-Macaulay binomial edge ideals

AbstractWe study the depth of classes of binomial edge ideals and classify all closed graphs whose binomial edge ideal is Cohen-Macaulay.

Binomial edge ideals and conditional independence statements

We introduce binomial edge ideals attached to a simple graph G and study their algebraic properties. We characterize those graphs for which the quadratic generators form a Gröbner basis in a lexicographic order induced by a vertex labeling. Such graphs are chordal and claw-free. We give a reduced squarefree Gröbner basis for general G. It follows that all binomial edge ideals are radical ideals. Their minimal primes can be characterized by particular subsets of the vertices of G. We provide sufficient conditions for Cohen–Macaulayness for closed and nonclosed graphs. Binomial edge ideals arise naturally in the study of conditional independence ideals. Our results apply for the class of conditional independence ideals where a fixed binary variable is independent of a collection of other variables, given the remaining ones. In this case the primary decomposition has a natural statistical interpretation.