Depth,Stanley depth,and regularity of ideals associated to graphs

Let \({\mathbb{K}}\) be a field and \({S=\mathbb{K}[x_1,\dots,x_n]}\) be the polynomial ring in n variables over \({\mathbb{K}}\). Let G be a graph with n vertices. Assume that \({I=I(G)}\) is the edge ideal of G and \({J=J(G)}\) is its cover ideal. We prove that \({{\rm sdepth}(J)\geq n-\nu_{o}(G)}\) and \({{\rm sdepth}(S/J)\geq n-\nu_{o}(G)-1}\), where \({\nu_{o}(G)}\) is the ordered matching number of G. We also prove the inequalities \({{\rmsdepth}(J^k)\geq {\rm depth}(J^k)}\) and \({{\rm sdepth}(S/J^k)\geq {\rmdepth}(S/J^k)}\), for every integer \({k\gg 0}\), when G is a bipartite graph. Moreover, we provide an elementary proof for the known inequality reg\({(S/I)\leq \nu_{o}(G)}\).     

The regularity of edge ideals of graphs

In this paper, we introduce some reduction processes on graphs which preserve the regularity of related edge ideals. Using these, an alternative proof for the theorem of R. Fröberg on the linearity of the resolutions of edge ideals is given.     

On residually ideals and projective dimension one modules

We prove that certain modules are faithful. This enables us to draw consequences about the reduction number and the integral closure of some classes of ideals.

     

Finite homological dimension and primes associated to integrally closed ideals

Let be an integrally closed ideal in a commutative Noetherian ring . Then the local ring is regular (resp. Gorenstein) for every if the projective dimension of is finite (resp. the Gorenstein dimension of is finite and satisfies Serre's condition (S)).

     

Regularity 3 in edge ideals associated to bipartite graphs

We focus in this paper on edge ideals associated to bipartite graphs and give a combinatorial characterization of those having regularity 3. When the regularity is strictly bigger than 3, we determine the first step i in the minimal graded free resolution where there exists a minimal generator of degree >i+3, show that at this step the highest degree of a minimal generator is i+4, and determine the corresponding graded Betti number β _i,i+4 in terms of the combinatorics of the graph. The results are then extended to the non-square-free case through polarization. We also study a family of ideals of regularity 4 that play an important role in our main result and whose graded Betti numbers can be completely described through closed combinatorial formulas.     

On the stability and regularity of the multiplier ideals of monomial ideals

Let a ? ?[x ₁, . . . , x _d ] be a monomial ideal and J (a) its multiplier ideal which is also a monomial ideal. It is proved that if a is strongly stable or squarefree strongly stable then so is J (a). Denote the maximal degree of minimal generators of a by d(a). When a is strongly stable or squarefree strongly stable, it is shown that the Castelnuovo-Mumford regularity of J (a) is less than or equal to d(a). As a corollary, one gets a vanishing result on the ideal sheaf]\(\widetilde {\mathcal{J}\left( a \right)}\) on ? ^d–1 associated to J (a) that H ⁱ(? ^d–1;\(\widetilde {\mathcal{J}\left( a \right)}\)(s–i)) = 0, for all i > 0 and s ≥ d(a).     

On the regularity of -Borel ideals

In this paper we prove Pardue's conjecture on the regularity of principal -Borel ideals. As a consequence we obtain an upper bound for the regularity of general -Borel 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 coincides with the regularity of 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.     

On the critical ideals of graphs

We introduce some determinantal ideals of the generalized Laplacian matrix associated to a digraph G, that we call critical ideals of G. Critical ideals generalize the critical group and the characteristic polynomials of the adjacency and Laplacian matrices of a digraph. The main results of this article are the determination of some minimal generator sets and the reduced Gröbner basis for the critical ideals of the complete graphs, the cycles and the paths. Also, we establish a bound between the number of trivial critical ideals and the stability and clique numbers of a graph.     

Ring graphs and toric ideals

We study the family of simple graphs whose number of primitive cycles equals its cycle rank. Then we study toric ideals of simple and oriented graphs.     

Characterization of projective graphs

We denote the distance between vertices x and y of a graph by d(x, y), and p_ij(x, y) = ∥ {z : d(x, z) = i, d(y, z) = j} ∥. The (s, q, d)-projective graph is the graph having the s-dimensional subspaces of a d-dimensional vector space over GF(q) as vertex set, and two vertices x, y adjacent iff $\text{dim}\text{(x ? y) = s ? 1}$. These graphs are regular graphs. Also, there exist integers λ and μ > 4 so that μ is a perfect square, p₁₁(x, y) = λ whenever d(x, y) = 1, and p₁₁(x, y) = μ whenever d(x, y) = 2. The (s, q, d)-projective graphs where $\text{2d}\text{3}\text{≤ s < d ? 2}$ and $\text{(s, q, d) ≠ (}\text{2d}\text{3}\text{, 2, d)}$, are characterized by the above conditions together with the property that there exists an integer r satisfying certain inequalities.     

Embedding of circulant graphs and generalized Petersen graphs on projective plane

Both the circulant graph and the generalized Petersen graph are important types of graphs in graph theory. In this paper, the structures of embeddings of circulant graph C(2n + 1; {1, n}) on the projective plane are described, the number of embeddings of C(2n + 1; {1, n}) on the projective plane follows, then the number of embeddings of the generalized Petersen graph P(2n +1, n) on the projective plane is deduced from that of C(2n +1; {1, n}), because C(2n + 1;{1, n}) is a minor of P(2n + 1, n), their structures of embeddings have relations. In the same way, the number of embeddings of the generalized Petersen graph P(2n, 2) on the projective plane is also obtained.     

Evaluations of initial ideals and Castelnuovo-Mumford regularity

This paper characterizes the Castelnuovo-Mumford regularity by evaluating the initial ideal with respect to the reverse lexicographic order.

     

Metrically and topologically projective ideals of Banach algebras

In the present paper, necessary conditions for the metric and topological projectivity of closed ideals of Banach algebras are given. In the case of commutative Banach algebras, a criterion for the metric and topological projectivity of ideals admitting a bounded approximate identity is obtained. The main result of the paper is as follows: a closed ideal of an arbitrary C*-algebra is metrically or topologically projective if and only if it admits a self-adjoint right identity.     

On Castelnuovo-Mumford regularity of projective curves

We give an effective method to compute the regularity of a saturated ideal defining a projective curve that also determines in which step of a minimal graded free resolution of the regularity is attained.