Ideal Graph Research Articles

Development of the whole-brain functional connectome explored via graph theory analysis

Introduction Adolescence is a period of remarkable development as children’s brains change to resemble adult brains. Resting state fMRI measures fluctuations in blood-oxygen signal from which we can infer functional connectivity (FC). Graph theory is a branch of mathematics that can quantify the complex patterns of connectivity and network architecture inherent in the functional connectome. An ideal graph theory analysis explores edges that are weighted, directional, and heterogenous (can be positive or negative). Recent developmental studies have applied graph theory to the functional connectome, yet due to the considerable complexity added by each facet, most ignore one or more aspects of an ideal graph theory analysis (directionality and heterogeneity). Methods The present cross-sectional study measured FC in typically developing children, adolescents, and young adults (age 6-24 years) using 150+ echo-planar volumes (3.6mm isotropic voxels, repetition/echo time=2000/30ms) acquired at rest. A standard pre-processing pipeline was used, and the functional connectome was quantified using a weighted, directed graph analysis, including both positive and negative connections. Five different graph theory metrics were utilized to quantify developmental trajectories: connection density, modularity, clustering coefficient, global efficiency, and betweenness centrality. Positive and negative connections were analyzed separately, and age and sex associations were explored. Results The total sample comprised 219 participants (mean age (SD) [range] = 14.1 (3.3) [6.5-24.0] years, 50% female). For positive connections, modularity and betweenness centrality increased with age (both p<0.001), while connection density, clustering coefficient, and global efficiency decreased with age (all p<0.001). By contrast, for negative connections, modularity and betweenness centrality decreased with age (p=0.002, p=0.003), while connection density, clustering coefficient , and global efficiency increased with age (p<0.001, p<0.001, p=0.003). Effects of sex, hemisphere, and their interaction were minimal, though global efficiency for negative connections was higher in the right hemisphere than the left (p<0.001). Conclusion Graph theory appears to be a useful tool for quantifying the complex development of the functional connectome. The developmental changes presented here may be driven by an intrinsic pressure to balance functionality with low metabolic cost to maintain the network. The positive connection network appears to shift towards a more efficient conformation resembling “small-world” architecture. In contrast, the negative connection network seems to shift away from such efficient architecture, possibly to prioritize improving functionality before later refinement.

BAB-GSL: Using Bayesian influence with attention mechanism to optimize graph structure in basic views

In recent years, Graph Neural Networks (GNNs) have garnered significant attention, with a notable focus on Graph Structure Learning (GSL), a branch dedicated to optimizing graph structures to enhance network training performance. Current GSL methods primarily involve constructing optimized graph representations by analyzing one or more initial graph sources to improve performance in subsequent application tasks. Despite these advancements, achieving high-quality graphs that accurately and robustly reflect node relationships remains challenging. This paper introduces a novel approach, termed BAB-GSL, designed to approximate an ideal graph structure through a systematic process. Specifically, two basic views are extracted from the original graph and utilized as inputs for the model, where the preliminary optimized view is generated through the view fusion module. The Attention mechanism is then applied to the optimized view to improve nodes’ connectivity and expressiveness. Subsequently, the trained view is re-structured using a Bayesian optimizer to produce the final graph structure. Extensive experiments were conducted across multiple datasets, both in undisturbed and attacked scenarios, to thoroughly evaluate the proposed method, demonstrating the effectiveness and robustness of the BAB-GSL approach.

On Energy of Prime Ideal Graph of a Commutative Ring Associated with Transmission-Based Matrices

This research explores the energy of the prime ideal graph of a commutative ring. The study demonstrates the energy formula of the graph associated with transmission-based matrices. Through research, the findings highlight the distance, Wiener-Hosoya, and distance signless Laplacian matrices. It should be noted that the distance and Wiener-Hosoya energies are always twice their spectral radius, meanwhile, it does not hold for distance signless Laplacian energy.

On Artinian Rings whose Ideal Graph is a Star

On Artinian Rings whose Ideal Graph is a Star

Counterexamples for some results in “On the module intersection graph of ideals of rings”

Counterexamples for some results in “On the module intersection graph of ideals of rings”

Strongly annihilator ideal graph of a lattice

For a lattice [Formula: see text], we associate a graph called the strongly annihilator ideal graph of [Formula: see text], which is denoted by [Formula: see text]. It is a graph with vertex set consists of all ideals of [Formula: see text] having nontrivial annihilators, and any two distinct vertices [Formula: see text] and [Formula: see text] are adjacent in [Formula: see text] if and only if the annihilator of [Formula: see text] contains a nonzero element of [Formula: see text] and the annihilator of [Formula: see text] contains a nonzero element of [Formula: see text]. We show that [Formula: see text] is connected with the diameter at most two and its girth is 3, 4, or infinity. We characterize all lattices whose [Formula: see text] is planar. Among other results, it is proved that [Formula: see text] is perfect.

Annihilator intersection graph of a lattice

For a lattice [Formula: see text], we associate a graph called the annihilator intersection graph of [Formula: see text], denoted by [Formula: see text] The vertex set of [Formula: see text] is the set of all nonzero zero-divisors of [Formula: see text] and any two distinct vertices [Formula: see text] and [Formula: see text] are adjacent in [Formula: see text] if and only if [Formula: see text]. It has shown that the [Formula: see text] is disconnected if and only if the number of atoms in [Formula: see text] is two. If [Formula: see text] is connected, then we determine the diameter and the girth of [Formula: see text] We characterize all lattices whose annihilator intersection graph is planar. Further, we obtain the clique number and chromatic number of [Formula: see text] when [Formula: see text] is a finite Boolean lattice. We show that the domination number of [Formula: see text] is not exceeding two. Finally, we obtain a condition under which the annihilator intersection graph is identical with the zero-divisor graph and the annihilator ideal graph of lattices.

Topological Indices and Properties of the Prime Ideal Graph of a Commutative Ring and its Line Graph

Let R be a commutative ring with identity, and P be a prime ideal of R. The prime ideal graph, denoted by Γp, is the graph where the set of vertices is R/{0} and two vertices are joined by an edge if their product belongs to P. This paper will discuss topological indices and some properties of the prime ideal graph of a commutative ring and its line graph. Topological indices, such as the Wiener, first Zagreb, second Zagreb, Harary, Gutman, Schultz, and Harmonic indices, are related to the degree of vertices and the diameter of the graphs. In this paper, we also discuss the independence and domination numbers of the line graph.

Topological Indices and Properties of the Prime Ideal Graph of a Commutative Ring and Its Line Graph

Let R be a commutative ring with identity, and P be a prime ideal of R. The prime ideal graph, denoted by Γp, is the graph where the set of vertices is R/{0} and two vertices are joined by an edge if their product belongs to P. This paper will discuss topological indices and some properties of the prime ideal graph of a commutative ring and its line graph. Topological indices, such as the Wiener, first Zagreb, second Zagreb, Harary, Gutman, Schultz, and Harmonic indices, are related to the degree of vertices and the diameter of the graphs. In this paper, we also discuss the independence and domination numbers of the line graph.

Eulerian and pancyclic zero-divisor graphs of ordered sets

In this paper, we determine when the zero-divisor graph of a special class of a finite pseudocomplemented poset is Eulerian. Also, we deal with Hamiltonian, vertex pancyclic, and edge pancyclic properties of the complement of a zero-divisor graph of finite 0-distributive posets. These results are applied to study the Eulerian, Hamiltonian and pancyclic properties of the zero-divisor graph, comaximal ideal graph, annihilating ideal graph, intersection ideal graph of a special class of commutative rings and intersection graph of a special class of groups.

Some results on a supergraph of the sum annihilating ideal graph of a commutative ring

The rings considered in this paper are commutative with identity which are not integral domains. Let [Formula: see text] be a ring. An ideal [Formula: see text] of [Formula: see text] is said to be an annihilating ideal if there exists [Formula: see text] such that [Formula: see text]. Let [Formula: see text] denote the set of all annihilating ideals of [Formula: see text] and we denote [Formula: see text] by [Formula: see text]. With [Formula: see text], in this paper, we associate an undirected graph denoted by [Formula: see text] whose vertex set is [Formula: see text] and two distinct vertices [Formula: see text] are adjacent in this graph if and only if either [Formula: see text] or [Formula: see text]. The aim of this paper is to study the interplay between some graph properties of [Formula: see text] and the algebraic properties of [Formula: see text] and to compare some graph properties of [Formula: see text] with the corresponding graph properties of the annihilating ideal graph of [Formula: see text] and the sum annihilating ideal graph of [Formula: see text].

Chordal and Perfect Zero-Divisor Graphs of Posets and Applications to Graphs Associated with Algebraic Structures

ABSTRACT In this paper, we characterize the perfect zero-divisor graphs and chordal zero-divisor graphs (its complement) of ordered sets. These results are applied to the zero-divisor graphs of finite reduced rings, the comaximal ideal graphs of rings, the annihilating ideal graphs of rings, the intersection graphs of ideals of rings, and the intersection graphs of subgroups of groups. In fact, it is shown that these graphs associated with a commutative ring R with identity can be effectively studied via the zero-divisor graph of a specially constructed poset from R.

ON PROPERTIES OF PRIME IDEAL GRAPHS OF COMMUTATIVE RINGS

The prime ideal graph of in a finite commutative ring with unity, denoted by , is a graph with elements of as its vertices and two elements in are adjacent if their product is in . In this paper, we explore some interesting properties of . We determined some properties of such as radius, diameter, degree of vertex, girth, clique number, chromatic number, independence number, and domination number. In addition to these properties, we study dimensions of prime ideal graphs, including metric dimension, local metric dimension, and partition dimension; furthermore, we examined topological indices such as atom bond connectivity index, Balaban index, Szeged index, and edge-Szeged index.

ANNIHILATING IDEAL AND EXACT ANNIHILATING IDEAL GRAPH OF RING Z_n

The existence of annihilator in the ring motivates the emergence of studies on Annihilating Ideal and Exact Annihilating Ideal Graphs. The purpose of this research is to describe the characteristics of an (exact) annihilating ideal of ring . The method used in this research is literature study. The results of this study discuss finiteness, adjacency, connectedness, vertices, and types of and . Furthermore, the number of vertices of an Annihilating Ideal Graph is determined by the factorization of . The adjacency of two vertices is determined by the divisibleness of . The results also show that is a subgraph of . can be represented as a union of several complete graphs.

On the Inclusion Ideal Graph of Semigroups

The inclusion ideal graph [Formula: see text] of a semigroup [Formula: see text] is an undirected simple graph whose vertices are all the nontrivial left ideals of [Formula: see text] and two distinct left ideals [Formula: see text], [Formula: see text] are adjacent if and only if either [Formula: see text] or [Formula: see text]. The purpose of this paper is to study algebraic properties of the semigroup [Formula: see text] as well as graph theoretic properties of [Formula: see text]. We investigate the connectedness of [Formula: see text] and show that the diameter of [Formula: see text] is at most 3 if it is connected. We also obtain a necessary and sufficient condition of [Formula: see text] such that the clique number of [Formula: see text] is the number of minimal left ideals of [Formula: see text]. Further, various graph invariants of [Formula: see text], viz. perfectness, planarity, girth, etc., are discussed. For a completely simple semigroup [Formula: see text], we investigate properties of [Formula: see text] including its independence number and matching number. Finally, we obtain the automorphism group of [Formula: see text].

On the intersection ideal graph of semigroups

The intersection ideal graph Γ(S) of a semigroup S is a simple undirected graph whose vertices are all nontrivial left ideals of S and two distinct left ideals I, J are adjacent if and only if their intersection is nontrivial. In this paper, we investigate the connectedness of Γ(S). We show that if Γ(S) is connected, then the diameter of Γ(S) is at most two. Further, we classify the semigroups S in terms of their ideals such that the diameter of Γ(S) is two. We obtain the domination number, independence number, girth and the strong metric dimension of Γ(S). We have also investigated the completeness, planarity and perfectness of Γ(S). We show that if S is a completely simple semigroup, then Γ(S) is weakly perfect. More over, in this article, we give an upper bound of the chromatic number of Γ(S). Finally, if S is the union of n minimal left ideals, then we obtain the metric dimension and the automorphism group of Γ(S).

On graphs with equal coprime index and clique number

Recently, Katre et al. introduced the concept of the coprime index of a graph. They asked to characterize the graphs for which the coprime index is the same as the clique number. In this paper, we partially solve this problem. In fact, we prove that the clique number and the coprime index of a zero-divisor graph of an ordered set and the zero-divisor graph of a ring Z p n coincide. Also, it is proved that the annihilating ideal graphs, the co-annihilating ideal graphs and the comaximal ideal graphs of commutative rings can be realized as the zero-divisor graphs of specially constructed posets. Hence the coprime index and the clique number coincide for these graphs as well.

Superfluous ideals of N-groups

We consider a right nearring N and a module over N (known as, N-group). For an arbitrary ideal (or N-subgroup) varOmega of an N-group G, we define the notions varOmega -superfluous, strictly varOmega -superfluous, g-superfluous ideals of G. We give suitable examples to distinguish between these classes and the existing classes studied in Bhavanari (Proc Japan Acad 61-A:23–25, 1985; Indian J Pure Appl Math 22:633–636, 1991; J Austral Math Soc 57:170–178, 1994), and prove some properties. For a zero-symmetric nearring with 1, we consider a module over a matrix nearring and obtain one-one correspondence between the superfluous ideals of an N-group (over itself) and those of M_{n}(N)-group N^{n}, where M_{n}(N) is the matrix nearring over N. Furthermore, we define a graph of superfluous ideals of a nearring and prove some properties with necessary examples.

Adjacency-like conditions and induced ideal graphs

In this paper we examine some natural ideal conditions and show how graphs can be defined that give a visualization of these conditions. We examine the interplay between the multiplicative ideal theory and the graph theoretic structure of the associated graph. Behavior of these graphs under standard ring extensions are studied, and in conjunction with the theory, some classical results and connections are made.

A generalization of the annihilating ideal graph for modules

A generalization of the annihilating ideal graph for modules