Related papers: On reduced zero-divisor graphs of posets
In this paper, we define a class of auxiliary graphs associated with simple undirected graphs. This class of auxiliary graphs is based on the set of spanning trees of the original graph and the edges constituting those spanning trees. A…
We introduce and study the problem of finding necessary and sufficient conditions under which a conformal blocks divisor on $\bar{M}_{0,n}$ is nonzero. We give necessary conditions in type A, which are sufficient when theta and critical…
A coprime labeling of a graph of order $n$ is an assignment of distinct positive integer labels in which adjacent vertices have relatively prime labels. Restricting labels to only the set $1$ to $n$ results in a prime labeling. In this…
There is an extensive recent literature on the graded, non-graded, prime, primitive, maximal ideals of Leavitt path algebras. In this introductory level survey, we will be giving an overview of different types of ideals and the…
If $I$ is a (two-sided) ideal of a ring $R$, we let $\operatorname{ann}_l(I)=\{r\in R\mid rI=0\},$ $\operatorname{ann}_r(I)=\{r\in R\mid Ir=0\},$ and $\operatorname{ann}(I)=\operatorname{ann}_l(I)\cap \operatorname{ann}_r(I)$ be the left,…
This article investigates the properties of order-divisor graphs associated with finite groups. An order-divisor graph of a finite group is an undirected graph in which the set of vertices includes all elements of the group, and two…
We provide some new conditions under which the graded Betti numbers of a monomial ideal can be computed in terms of the graded Betti numbers of smaller ideals, thus complementing Eliahou and Kervaire's splitting approach. As applications,…
In combinatorial commutative algebra and algebraic statistics many toric ideals are constructed from graphs. Keeping the categorical structure of graphs in mind we give previous results a more functorial context and generalize them by…
We explore various techniques for counting the number of straight-edge crossing-free graphs that can be embedded on a planar point set. In particular, we derive a lower bound on the ratio of the number of such graphs with $m+1$ edges to the…
For a commutative ring $R$, the zero-divisor graph of $R$ is a simple graph with the vertex set as the set of all zero-divisors of $R$ and two distinct vertices $x$ and $y$ are adjacent if and only if $xy = 0$. This article attempts to…
On a metric graph we introduce the notion of a free divisor as a replacement for the notion of a base point free complete linear system on a curve. By means of an example we show that the Clifford inequality is the only obstruction for the…
The metric dimension of non-component graph, associated to a finite vector space, is determined. It is proved that the exchange property holds for resolving sets of the graph, except a special case. Some results are also related to an…
Let p be a singular point of a variety. Consider a resolution where the preimage of p is a simple normal crossing divisor E. The combinatorial structure of E is described by a cell complex D(E), called the dual graph or dual complex of E.…
We have tried to translate some graph properties of AG(R) and Gamma(R) to the topological properties of Zariski topology. We prove that Rad(Gamma(R)) and Rad(AG(R)) are equal and they are equal to 3, if and only if the zero ideal of R is an…
In this note we show that the initial ideal of the annihilator ideal of a generic form is generated by the largest possible monomials in each degree. We also show that the initial ideal with respect to the degree reverse lexicographical…
We investigate resolutions of letterplace ideals of posets. We develop topological results to compute their multigraded Betti numbers, and to give structural results on these Betti numbers. If the poset is a union of no more than $c$…
We give a combinatorial characterization of upward planar graphs in terms of upward planar orders, which are special linear extensions of edge posets.
For many types of graphs, criteria have been discovered that give necessary and sufficient conditions for an integer sequence to be the degree sequence of such a graph. These criteria tend to take the form of a set of inequalities, and in…
In this paper we construct a class of bounded degree bipartite graphs with a small separator and large bandwidth. Furthermore, we also prove that graphs from this class are spanning subgraphs of graphs with minimum degree just slightly…
This note describes necessary and sufficient conditions for a sequence of positive integers to be the degree sequence of a connected simple graph. Conditions are also given under which a sequence is necessarily connected i.e. the sequence…