Related papers: Gorenstein binomial edge ideals associated with sc…

We introduce a class of ideals generated by a set of 2-minors of $m\times n$-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…

Let $\Delta$ be a 1-dimensional simplicial complex. Then $\Delta$ may be identified with a finite simple graph $G$. In this article, we investigate the toric ring $R_G$ of $G$. All graphs $G$ such that $R_G$ is a normal domain are…

Let $J_G$ denote the binomial edge ideal of a connected undirected graph on $n$ vertices. This is the ideal generated by the binomials $x_iy_j - x_jy_i, 1\leq i < j \leq n,$ in the polynomial ring $S= K[x_1,...,x_n,y_1,...,y_n]$ where…

A central question in liaison theory asks whether every Cohen-Macaulay, graded ideal of a standard graded K-algebra belongs to the same G-liaison class of a complete intersection. In this paper we answer this question positively for toric…

We investigate, using the notion of linear quotients, significative classes of connected graphs whose monomial edge ideals, not necessarily squarefree, have linear resolution, in order to compute standard algebraic invariants of the…

Let $G$ be a Cameron--Walker graph on $n$ vertices and $J_G$ the binomial edge ideal of $G$. Let $S$ denote the polynomial ring in $2n$ variables over a field. It is shown that the following conditions are equivalent: (i) $S/J_G$ is…

We study powers of binomial edge ideals associated with closed and block graphs.

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…

When $I$ is the edge ideal of a graph $G$, we use combinatorial properities, particularly Property $P$ on connectivity of neighbors of an edge, to classify when a binomial sum of vertices is a regular element on $R/I(G)$. Under a mild…

Let G be a perfect graph and let J be its ideal of vertex covers. We show that the Rees algebra of J is normal and that this algebra is Gorenstein if G is unmixed. Then we give a description--in terms of cliques--of the symbolic Rees…

In this paper we give upper bounds for the regularity of edge ideal of some classes of graphs in terms of invariants of graph. We introduce two numbers $a'(G)$ and $n(G)$ depending on graph $G$ and show that for a vertex decomposable graph…

In this paper we prove the conjectured upper bound for Castelnuovo-Mumford regularity of binomial edge ideals posed in [23], in the case of chordal graphs. Indeed, we show that the regularity of any chordal graph G is bounded above by the…

Closed graphs have been characterized by Herzog et al. as the graphs whose binomial edge ideals have a quadratic Gr\"obner basis with respect to a diagonal term order. In this paper, we focus on a generalization of closed graphs, namely…

This research focuses on analyzing the depth of generalized binomial edge ideals. We extend the notion of $d$-compatible map for the pairs of a complete graph and an arbitrary graph, and using it, we give a combinatorial lower bound for the…

The regularity of an edge ideal of a finite simple graph $G$ is at least the induced matching number of $G$ and is at most the minimum matching number of $G$. If $G$ possesses a dominating inuduced matching, i.e., an induced matching which…

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 Gr\"obner basis in a lexicographic order induced by a vertex…

An integral convex polytope ${\mathcal P}$ is said to be Gorenstein if its toric ring $K[{\mathcal P}]$ is normal and Gorenstein. In this paper, Gorenstein cut polytopes of graphs are characterized explicitly. First, we prove that…

Let $(A,\mathfrak m)$ be a two-dimensional excellent normal Gorenstein local domain containing an algebraically closed filed. Let $I =H^0(X,\mathcal{O}_X(-Z)) \subset A$ be an $\mathfrak m$-primary integrally closed ideal represented by an…

In this article, we obtain an upper bound for the regularity of the binomial edge ideal of a graph whose every block is either a cycle or a clique. As a consequence, we obtain an upper bound for the regularity of binomial edge ideal of a…

In this paper, we introduce generalized Gorenstein local (GGL) rings. The notion of GGL rings is a natural generalization of the notion of almost Gorenstein rings, which can thus be treated as part of the theory of GGL rings. For a…