Ideal Graph Research Articles

The annihilating-ideal graph of commutative semigroups

In this paper, we associate an undirected graph AG(S), the annihilating-ideal graph, to a commutative semigroup S. This graph has vertex set A⁎(S)=A(S)∖{(0)}, where A(S) is the set of proper ideals of S with nonzero annihilator. Two distinct vertices I,J∈A⁎(S) are defined to be adjacent in AG(S) if and only if IJ=(0), the zero ideal. Conditions are given to ensure a finite graph. Semigroups for which each nonzero, proper ideal of S is an element of A⁎(S) are characterized. Connections are drawn between AG(S) and Γ(S), the well-known zero-divisor graph, and the connectivity, diameter, and girth of AG(S) are described. Semigroups S for which AG(S) is a complete or star graph are characterized. Finally, it is proven that the chromatic number is equal to the clique number of the annihilating ideal graph for each reduced semigroup and null semigroup. Upper and lower bounds for χ(AG(S)) are given for a general commutative semigroup.

Co-maximal ideal graphs of matrix algebras

Let R be a ring with unity. The co-maximal ideal graph of R, denoted by \(\Gamma (R)\), is a graph whose vertices are all non-trivial left ideals of R, and two distinct vertices \(I_1\) and \(I_2\) are adjacent if and only if \(I_1 + I_2 = R\). In this paper, some results on the co-maximal ideal graphs of matrix algebras are given. For instance, we determine the domination number, the clique number and a lower bound of the independence number of \(\Gamma (M_n(\mathbb {F}_q))\), where \(M_n(\mathbb {F}_q)\) is the ring of \(n\times n\) matrices over the finite field \(\mathbb {F}_q\). Furthermore, we characterize all rings (not necessarily commutative) whose domination numbers of their co-maximal ideal graphs are finite. Among other results, we show that if \(\Gamma (R)\cong \Gamma (M_n(\mathbb {F}_q))\), where \(n\ge 2\) is a positive integer and R is a ring, then \(R\cong M_n(\mathbb {F}_q)\). Also, it is proved that if R and \(R'\) are two finite reduced rings and \(\Gamma (M_m(R))\cong \Gamma (M_n(R'))\), for some positive integers \(m,n\ge 2\), then \(m=n\) and \(R\cong R'\).

Ideal Whitehead graphs in Out(Fr) III: Achieved graphs in rank 3

By proving precisely which singularity index lists arise from the pair of invariant foliations for a pseudo-Anosov surface homeomorphism, Masur and Smillie [24] determined a Teichmüller flow invariant stratification of the space of quadratic differentials. In this final paper of a three-paper series, we give a first step to an [Formula: see text] analog of the Masur–Smillie theorem. Since the ideal Whitehead graphs defined by Handel and Mosher [16] give a strictly finer invariant in the analogous [Formula: see text] setting, we determine which of the 21 connected, simplicial, five-vertex graphs are ideal Whitehead graphs of fully irreducible outer automorphisms in [Formula: see text]. The rank 2 case is actually a direct consequence of the work of Masur and Smillie, as all elements of [Formula: see text] are induced by surface homeomorphisms and the index list and ideal Whitehead graph for a surface homeomorphism give precisely the same data.

When is the annihilating ideal graph of a zero-dimensional semiquasilocal commutative ring planar? Nonquasilocal case

The rings considered in this article are commutative with identity and which are not integral domains. The aim of this article is to classify zero-dimensional semiquasilocal rings with at least two maximal ideals such that their annihilating ideal graphs are planar.

On the dominating sets of the complement of the annihilating ideal graph of a commutative ring

Let R be a commutative ring with identity which is not an integral domain. Let A(R)∗ denote the collection of all nonzero annihilating ideals of R and AG(R) denote the annihilating ideal graph of R. In this article, we consider the dominating sets of (AG(R))c (where (AG(R))c is the complement of AG(R)) and study the influence of the dominating sets of (AG(R))c on the ring structure of R and vice-versa.

Graph-without-cut: An Ideal Graph Learning for Image Segmentation

Graph-based image segmentation organizes the image elements into graphs and partitions an image based on the graph. It has been widely used and many promising results are obtained. Since the segmentation performance highly depends on the graph, most of existing methods focus on obtaining a precise similarity graph or on designing efficient cutting/merging strategies. However, these two components are often conducted in two separated steps, and thus the obtained graph similarity may not be the optimal one for segmentation and this may lead to suboptimal results. In this paper, we propose a novel framework, Graph-Without-Cut (GWC), for learning the similarity graph and image segmentations simultaneously. GWC learns the similarity graph by assigning adaptive and optimal neighbors to each vertex based on the spatial and visual information. Meanwhile, the new rank constraint is imposed to the Laplacian matrix of the similarity graph, such that the connected components in the resulted similarity graph are exactly equal to the region number. Extensive empirical results on three public data sets (i.e, BSDS300, BSDS500 and MSRC) show that our unsupervised GWC achieves state-of-the-art performance compared with supervised and unsupervised image segmentation approaches.

On the comaximal ideal graph of a commutative ring

Let $R$ be a commutative ring with identity. We use $\Gamma ( R )$ to denote the comaximal ideal graph. The vertices of $\Gamma ( R )$ are proper ideals of R that are not contained in the Jacobson radical of $R$, and two vertices $I$ and $J$ are adjacent if and only if $I + J = R$. In this paper we show some properties of this graph together with the planarity and perfection of $\Gamma ( R )$.

Some results on the complement of the annihilating ideal graph of a commutative ring

Rings considered in this article are commutative with identity which admit at least one nonzero annihilating ideal. For such a ring R, we determine necessary and sufficient conditions in order that the complement of its annihilating ideal graph is connected and also find its diameter when it is connected. We discuss the girth of the complement of the annihilating ideal graph of R and prove that it is either equal to 3 or ∞. We also present a necessary and sufficient condition for the complement of the annihilating ideal graph to be complemented.

On the clique number of the complement of the annihilating ideal graph of a commutative ring

The rings considered in this article are commutative with identity which admit at least one nonzero annihilating ideal. Let \(R\) be such a ring. If \(Z(R)\), the set of zero-divisors of \(R\), is not an ideal of \(R\), then it is shown in this article that the complement of the annihilating ideal graph of \(R\) does not contain any infinite clique if and only if its clique number is finite. Moreover, this article classifies (up to isomorphism of rings) rings \(R\) such that \(Z(R)\) is not an ideal and for which the complement of its annihilating ideal graph does not admit any infinite clique. Furthermore, the clique number of the complement of the annihilating ideal graph of \(R\) is determined under the assumption that \(R\) is reduced with the property that the complement of its annihilating ideal graph does not contain any infinite clique. In addition, this article also considers rings \(R\) such that \(Z(R)\) is an ideal of \(R\) and some necessary conditions are determined in order that the complement of the annihilating ideal graph of \(R\) does not contain any infinite clique. Moreover, for a chained ring \(R\), it is shown that the complement of the annihilating ideal graph of \(R\) does not contain any infinite clique if and only if its clique number is finite. The clique number of the complement of the annihilating ideal graph of a chained ring \(R\) is determined under the hypothesis that it does not contain any infinite clique.

The Inclusion Ideal Graph of Rings

Let R be a ring with unity. The inclusion ideal graph of a ring R, denoted by In(R), is a graph whose vertices are all nontrivial left ideals of R and two distinct left ideals I and J are adjacent if and only if I ⊂ J or J ⊂ I. In this paper, we show that In(R) is not connected if and only if R ≅ M 2(D) or D 1 × D 2, for some division rings, D, D 1 and D 2. Moreover, we prove that if In(R) is connected, then diam(In(R)) ≤3. It is shown that if In(R) is a tree, then In(R) is a caterpillar with diam(In(R)) ≤3. Also, we prove that the girth of In(R) belongs to the set {3, 6, ∞}. Finally, we determine the clique number and the chromatic number of the inclusion ideal graph for some classes of rings.

Complemented graphs and blow-ups of Boolean graphs, with applications to co-maximal ideal graphs

For a set X, let 2X be the power set of X. Let BX be the Boolean graph, which is defined on the vertex set 2X \ {X, ?}, with M adjacent to N if M ? N = ?. In this paper, several purely graph-theoretic characterizations are provided for blow-ups of a finite or an infinite Boolean graph (respectively, a preatomic graph). Then the characterizations are used to study co-maximal ideal graphs that are blow-ups of Boolean graphs (pre-atomic graphs, respectively).

Graph properties of co-maximal ideal graphs of commutative rings

In this paper, we determine the diameters of graphs [Formula: see text] and [Formula: see text] for a ring R with infinitely many maximal ideals. We also use graph blow-up to give a complete classification of rings R whose graphs [Formula: see text] are non-empty planar graphs.

When is the annihilating ideal graph of a zero-dimensional quasisemilocal commutative ring complemented?

Let R be a commutative ring with identity. Let A(R) denote the collection of all annihilating ideals of R (that is, A(R) is the collection of all ideals I of R which admits a nonzero annihilator in R). Let AG(R) denote the annihilating ideal graph of R. In this article, necessary and sufficient conditions are determined in order that AG(R) is complemented under the assumption that R is a zero-dimensional quasisemilocal ring which admits at least two nonzero annihilating ideals and as a corollary we determine finite rings R such that AG(R) is complemented under the assumption that A(R) contains at least two nonzero ideals.

The intersection graph of ideals of a poset

Let (P, ≤) be a partially ordered set (poset, briefly) with a least element 0. The intersection graph of ideal of P, denoted by G(P), is a graph whose vertices are all non-trivial ideals of P and two distinct vertices I and J are adjacent if and only if I ∩ J ≠ {0}. In this paper, we study some relations between the algebraic properties of posets and graph-theoretic properties of G(P). We investigate the connectivity, diameter, girth and planarity of the intersection graph. Also, among the other things, we show that if the clique number of G(P) is finite, then P is finite too.

ON THE GENUS OF THE INTERSECTION GRAPH OF IDEALS OF A COMMUTATIVE RING

To each commutative ring R one can associate the graph G(R), called the intersection graph of ideals, whose vertices are nontrivial ideals of R. In this paper, we try to establish some connections between commutative ring theory and graph theory, by study of the genus of the intersection graph of ideals. We classify all graphs of genus 2 that are intersection graphs of ideals of some commutative rings and obtain some lower bounds for the genus of the intersection graph of ideals of a nonlocal commutative ring.

Implements of Graph Blow-Up in Co-Maximal Ideal Graphs

In this article, graph blow-up is used to study the implements of the co-maximal graph and the co-maximal ideal graph 𝒞(R) of a commutative ring R. We show that for a ring R with finitely many maximal ideals, the graph 𝒞(R) is a blow-up of a Boolean graph, and we give more detailed structure of the graphs and 𝒞(R) as blow-ups of Boolean graphs. We determine for what rings R, 𝒞(R) or is a finite graph, respectively. In the end, we get some new results for rings by taking advantage of the graph .

On the inclusion ideal graph of a ring

The inclusion ideal graph of a ring R, denoted by In(R), is a graph whose vertices are all non-trivial left ideals of R and two distinct left ideals I and J are adjacent if and only if I⊆J or J⊆I. In this paper, we show that In(R) is not connected if and only if R≅M2(D) or D1×D2, for some division rings, D,D1 and D2. Moreover, if R is connected, then diam(In(R))⩽3. We prove that if In(R) is a tree, then In(R) is a star graph or P4. Also, In(R) is a complete graph if and only if R is a uniserial ring. Next, it is shown that the inclusion ideal graph of Mn(D) for a division ring D and a natural number n>3 is not regular.

A Graph Lattice Approach to Maintaining and Learning Dense Collections of Subgraphs as Image Features

Effective object and scene classification and indexing depend on extraction of informative image features. This paper shows how large families of complex image features in the form of subgraphs can be built out of simpler ones through construction of a graph lattice—a hierarchy of related subgraphs linked in a lattice. Robustness is achieved by matching many overlapping and redundant subgraphs, which allows the use of inexpensive exact graph matching, instead of relying on expensive error-tolerant graph matching to a minimal set of ideal model graphs. Efficiency in exact matching is gained by exploitation of the graph lattice data structure. Additionally, the graph lattice enables methods for adaptively growing a feature space of subgraphs tailored to observed data. We develop the approach in the domain of rectilinear line art, specifically for the practical problem of document forms recognition. We are especially interested in methods that require only one or very few labeled training examples per category. We demonstrate two approaches to using the subgraph features for this purpose. Using a bag-of-words feature vector we achieve essentially single-instance learning on a benchmark forms database, following an unsupervised clustering stage. Further performance gains are achieved on a more difficult dataset using a feature voting method and feature selection procedure.

Dimensionality reduction with adaptive graph

Graph-based dimensionality reduction (DR) methods have been applied successfully in many practical problems, such as face recognition, where graphs play a crucial role in modeling the data distribution or structure. However, the ideal graph is, in practice, difficult to discover. Usually, one needs to construct graph empirically according to various motivations, priors, or assumptions; this is independent of the subsequent DR mapping calculation. Different from the previous works, in this paper, we attempt to learn a graph closely linked with the DR process, and propose an algorithm called dimensionality reduction with adaptive graph (DRAG), whose idea is to, during seeking projection matrix, simultaneously learn a graph in the neighborhood of a prespecified one. Moreover, the pre-specified graph is treated as a noisy observation of the ideal one, and the square Frobenius divergence is used to measure their difference in the objective function. As a result, we achieve an elegant graph update formula which naturally fuses the original and transformed data information. In particular, the optimal graph is shown to be a weighted sum of the pre-defined graph in the original space and a new graph depending on transformed space. Empirical results on several face datasets demonstrate the effectiveness of the proposed algorithm.

Generating and verifying graph states for fault-tolerant topological measurement-based quantum computing in two-dimensional optical lattices

We propose two schemes for implementing graph states useful for fault-tolerant topological measurement-based quantum computation in 2D optical lattices. We show that bilayer cluster and surface code states can be created by global single-row and controlled-Z operations. The schemes benefit from the accessibility of atom addressing on 2D optical lattices and the existence of an efficient verification protocol which allows us to ensure the experimental feasibility of measuring the fidelity of the system against the ideal graph state. The simulation results show potential for a physical realization toward fault-tolerant measurement-based quantum computation against dephasing and unitary phase errors in optical lattices.