Related papers: Divisors on graphs, Connected flags, and Syzygies

We present two new problems on lower bounds for resolution Betti numbers of monomial ideals generated in a fixed degree. The first concerns any such ideal and bounds the total Betti numbers, while the second concerns ideals that are…

Given a graph $G$, the \textit{independence complex} $I(G)$ is the simplicial complex whose faces are the independent sets of $V(G)$. Let $\tilde{b}_i$ denote the $i$-th reduced Betti number of $I(G)$, and let $b(G)$ denote the sum of…

In the field of algebraic systems biology, the number of minimal polynomial models constructed using discretized data from an underlying system is related to the number of distinct reduced Gr\"obner bases for the ideal of the data points.…

For a graph $G$, Postnikov-Shapiro \cite{PS04} construct two ideals $I_G$ and $J_G.$ $I_G$ is a monomial ideal and $J_G$ is generated by powers of linear forms. They proved the equality of their Hilbert series and conjectured that the…

Let $G$ be a simple graph on the vertex set $[n]$ and $J_G$ be the corresponding binomial edge ideal. Let $G=v*H$ be the cone of $v$ on $H$. In this article, we compute all the Betti numbers of $J_G$ in terms of Betti number of $J_H$ and as…

Minimal separators in graphs are an important concept in algorithmic graph theory. In particular, many problems that are NP-hard for general graphs are known to become polynomial-time solvable for classes of graphs with a polynomially…

We study homological properties of random quadratic monomial ideals in a polynomial ring $R = {\mathbb K}[x_1, \dots x_n]$, utilizing methods from the Erd\"{o}s-R\'{e}nyi model of random graphs. Here for a graph $G \sim G(n, p)$ we consider…

In this paper we introduce a working generalization of the theory of Gr\"obner bases for algebras of partial difference polynomials with constant coefficients. One obtains symbolic (formal) computation for systems of linear or non-linear…

Let $R = k[x_1, \dotsc , x_n]$ denote the standard graded polynomial ring over a field $k$. We study certain classes of equigenerated monomial ideals with the property that the so-called complementary ideal has no linear relations on the…

Describing families of ideals that are minimally generated by at least one, or by all, of their reduced Gr\"obner bases is a central topic in commutative algebra. In this paper, we address this problem in the context of toric ideals of…

In this paper we initiate the study of the total zero-divisor graphs over commutative rings with unity. These graphs are constructed by both relations that arise from the zero-divisor graph and from the total graph of a ring. We…

We use a Gr\"obner basis technique first introduced by Knutson, Miller and Yong to study the interplay between properties of a graph $G$ and algebraic properties of the toric ideal that it defines. We first recover a well-known height…

In this paper we will define analogs of Gr\"obner bases for $R$-subalgebras and their ideals in a polynomial ring $R[x_1,\ldots,x_n]$ where $R$ is a noetherian integral domain with multiplicative identity and in which we can determine ideal…

Edge ideals of finite simple graphs $G$ on $n$ vertices are the ideals $I(G)$ of the polynomial ring $S$ in $n$ variables generated by the quadratic monomials associated with the edges of $G$. In this paper, we consider the possible pairs…

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 \pi(G) denote the set of prime divisors of the order of a finite group G. The prime graph of G is the graph with vertex set \pi(G) with edges {p,q} if and only if there exists an element of order pq in G. In this paper, we prove that a…

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 $r\geq 3$ be an integer and $G$ be a graph. Let $\delta(G), \Delta(G)$, $\alpha(G)$ and $\mu(G)$ denotes minimum degree, maximum degree, independence number and matching number of $G$, respectively. Recently, Caro, Davila and Pepper…

Hilbert schemes of zero-dimensional ideals in a polynomial ring can be covered with suitable affine open subschemes whose construction is achieved using border bases. Moreover, border bases have proved to be an excellent tool for describing…

Several algebraic properties of a binomial edge ideal $J_G$ can be interpreted in terms of combinatorial properties of its associated graph $G$. In particular, the so-called cut sets of a graph $G$, special sets of vertices that disconnect…