Various Types Of Graphs Research Articles

Subring in Graph Theory

In our study, we delved into the intricacies of graph theory by exploring the properties of subrings within various types of graphs. By focusing on prime graphs and simple graphs, we unraveled the complex relationship between subring-prime graphs. Additionally, we delved into the concept of homomorphism within both simple subring graphs and prime subring graphs, adding depth to our analysis.

Solitonic solutions and stability analysis of Benjamin Bona Mahony Burger equation using two versatile techniques

This study aims to explore the precise resolution of the nonlinear Benjamin Bona Mahony Burgers (BBMB) equation, which finds application in a variety of nonlinear scientific disciplines including fluid dynamics, shock generation, wave transmission, and soliton theory. Within this paper, we employ two versatile methodologies, specifically the extended exp(-Ψ(χ))\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\exp (-\\Psi (\\chi ))$$\\end{document} expansion technique and the novel Kudryashov method, to identify the exact soliton solutions of the nonlinear BBMB equation. The solutions we discovered involve trigonometric functions, hyperbolic functions, and rational functions. The uniqueness of this research lies in uncovering the bright soliton, kink wave solution, and periodic wave solution, and conducting stability analysis. Furthermore, the solutions’ graphical characteristics were explored through the utilization of the mathematical software Maple 2022 (https://maplesoft.com/downloads/selectplatform.aspx?hash=61ab59890f2313b2241fde3423fd975e). The system’s physical interpretation is defined through various types of graphs, including contour graphs, 3D-surface graphs, and line graphs, which use appropriate parameter values. These recommended techniques hold significant importance and are applicable in diverse nonlinear evolutionary equations found in the field of nonlinear sciences for illustrating nonlinear physical models.

Lict k-Domination in Graphs

Objectives: In this study, we find lict -domination number of various types of graphs. Methods: Let be any graph, a set is said to be an -dominating set of lict graph if every vertex is dominated by at least vertices in , that is . The Lict -domination number is the minimum cardinality of -dominating set of . Findings: This study is centered on the lict -domination number of the graph and developed its relationship with other different domination parameters. Novelty: This study introduces the concept of “Lict -Domination in Graphs”. It obtains many bounds on in terms of vertices, edges, and other different parameters of . Keywords: Domination number, k-domination number, Lict graph, Lict k-domination number, Total domination number, Independent domination number

FEATURES OF THE APPLICATION OF THE SCILAB MATHEMATICAL PACKAGE IN THE PROCESS OF TEACHING THE DISCIPLINE «HIGHER MATHEMATICS»

The article notes that the choice of software is of great importance in teaching the discipline «Higher Mathematics». The complexity and type of tasks to be solved depend on the functionality of the selected program. It is emphasised that the methods of approximate, primarily numerical, solution are used. The methods of numerical solution of mathematical problems have always been an integral part of mathematics and have always been part of the content of natural-mathematical and engineering education. This article uses the Skilab math package as software. It is a multi-functional package designed for performing engineering and scientific calculations, which allows you to perform complex algebraic calculations, solve problems of differentiation and integration. All this contributes to the successful use of the Scilab mathematical package in the process of teaching the discipline «Higher Mathematic». The author draws attention to the fact that Skilab is a computer mathematics system designed to perform engineering and scientific calculations, such as: solving nonlinear equations and systems; solving linear algebra problems; solving optimization tasks; differentiation and integration; processing of experimental data (interpolation and approximation, method of least squares); solution of ordinary differential equations and systems. Skilab provides wide opportunities for creating and editing various types of graphs and surfaces. The Skilab system contains a sufficient number of built-in commands, operators and functions, and its distinctive feature is flexibility. The user can create any new command or function and then use it alongside the built-in ones. The article states that the use of modern mathematical packages significantly increases the intensity of cognitive activity, raises the level of mathematical preparation, improves the knowledge control system, and promotes learning motivation. The perspective of the development of the study of the use of mathematical packages is the development of a database on the issues of their use during the educational process, which should take into account the professional orientation of future activities. This approach provides a more effective application of interdisciplinary connections, which contributes to the in-depth study of the material and the expansion of opportunities for independent learning.

A Robust and Generalized Framework for Adversarial Graph Embedding

Graph embedding is essential for graph mining tasks. With the prevalence of graph data in real-world applications, many methods have been proposed in recent years to learn high-quality graph embedding for various types of graphs, among which the Generative Adversarial Networks (GAN) based methods attract increasing attention among researchers. However, most GAN-based generator-discriminator frameworks randomly generate the negative samples from the original graph distributions to enhance the training process of the discriminator without considering the noise. In addition, most of these methods only focus on the explicit graph structures and cannot fully capture complex semantics of edges such as various relationships or asymmetry. In order to address these issues, we propose a robust and generalized framework named AGE. It generates fake neighbors as the enhanced negative samples from the implicit distribution, and enables the discriminator and generator to jointly learn robust and generalized node representations. Based on this framework, we propose three models to handle three types of graph data and derive the corresponding optimization algorithms, namely the UG-AGE and DG-AGE for undirected and directed homogeneous graphs, respectively, and the HIN-AGE for heterogeneous information networks. Extensive experiments show that our methods consistently and significantly outperform existing state-of-the-art methods across multiple graph mining tasks.

DESIGN OF PID STRUCTURED TCSC CONTROLLER FOR SMALL SIGNAL STABILITY IMPROVEMENT USING GRAY WOLF OPTIMIZATION ALGORITHM

In power system stability plays a very important role and necessary to maintain the stability of system when different contingency applied. The Power system is very complex and interconnected this resulting into lower frequency oscillations due to variation in load which creates stability problems. Stability enhance by various controller and optimization technique. To eliminate the stability problems proposed PID structured TCSC controller used in SMIB system. Proposed work implements with MATLAB/ Simulink environment and the performance of the PID structured TCSC controller is compared for without and with GWO algorithm. Observations and results tested on different loading conditions as nominal, light and heavy loading. Different types of fault consider in different loading conditions and various types of graphs presented as speed deviation, power angle deviation etc. All contingencies tabulated and compare results without and with GWO algorithm. The system evaluation is done by time domain simulation using MATLAB/Simulink software. It is detected that the out of all algorithms GWO algorithm found more compatible for parameters reduces settling time and provide superior performance of the power system. Different simulation results are also obtained for various loading & fault conditions.

Reversibility of link prediction and its application to epidemic mitigation

Current link prediction strategies are about finding new probable strong relations to establish or weak ones to remove. An interesting strategy is utilizing link prediction to prioritize the edges in the network and finding newly probable established relations. In this paper we will introduce and explain RLP, reverse link prediction, as a new paradigm, and use popular basic scoring methods including CN, JC, AA, RA, and PA, as its core to examine. The test cases are nine datasets. Half of them are contact networks in different levels from personal contact to aviation, and another half is for covering different test situations. After reviewing the edge removal based epidemic mitigation methods, we show that RLP can be used to decrease the epidemics spreading speed as a general method with various link prediction algorithms, and here in this paper, preferential attachment (PA) has the best results overall. But the results heavily depend on the nature of the examined networks: regular, scale-free or small-world. We also propose an easy to understand criteria, path count, for comparing the efficacy of epidemics mitigation methods. RLP can be extended to use other link prediction scoring methods in various types of graphs as well.

Chromatic Number of Amalgamation of Wheel Graph-Star Graph and Amalgamation of Wheel Graph-Sikel Graph

<p>A graph denoted by is a pair of where is a non-empty set of vertices in G, and E is a set of edges in G. In graph theory, there are various types of graphs including star graphs, cycle graph, and wheel graph. Graph operations on two or more types of graphs can produce new graphs. Amalgamation is one of the operations on graphs. Suppose and are two connected graphs, amalgamating the vertices of graph 1 and graph 2 by joining vertices and vertices denoted by is the graph obtained by joins vertex of graph and vertex of graph into one vertex , where is the common vertex of the graph resulting from , while amalgamating the edges of graph and graph by joining edges and edge , denoted by are graphs obtained by combining edge of graph 1 and edge of graph 2 into one edge g, where g is common side of graph of (𝐺1,𝐺2;𝑒,𝑓). Vertex coloring is one of the fastest-growing topics in graphs. The coloring of the vertices, that is, assigning a color to each vertex of the graph so that each neighboring vertex has a different color. The minimum number of colors used to color the vertices of a graph is called the chromatic number. In this article, we discuss the chromatic number in the amalgamated graph of wheel graph and star graph and the chromatic number on the amalgamation of wheel graph and cycle graph. The results obtained are as follows: (1) the chromatic number of the amalgamated graph of the wheel graph () and the star graph () for n even is 3 and for n odd is 4, (2) the chromatic number of the graph resulting from the operation amalgamation of the wheel graph () and the cycle graph () for n even is 3 and for odd is 4.</p>

Graph Convolutional Networks Guided by Explicitly Estimated Homophily and Heterophily Degree

Graph convolutional networks (GCNs) have been successfully applied to learning tasks on graph-structured data. However, most traditional GCNs based on graph convolutions assume homophily in graphs, which leads to a poor performance when dealing with heterophilic graphs. Although many novel methods have recently been proposed to deal with heterophily, the effect of homophily and heterophily on classifying node pairs is not clearly separated in existing approaches and inevitably influences each other. To deal with various types of graphs more accurately, in this work we propose a new GCN-based model that leverages the explicitly estimated homophily and heterophily degree between node pairs and adaptively guides the propagation and aggregation of signed messages. We also design a pre-training process to learn homophily and heterophily degree from both original node attributes that are graph-agnostic and the localized graph structure information by using Deepwalk that reflects graph topology. Extensive experiments on eight real-world benchmarks demonstrate that the new approach achieves state-of-the-art results on three homophilic graph datasets and outperforms baselines on five heterophilic graph datasets.

Chaotic dynamics of a plankton-fish system with fear and its carry over effects in the presence of a discrete delay

Plankton-fish interactions are the central topic of interest related to marine ecology. Apart from the direct predation (lethal effect) of zooplankton by fish, there are some non-lethal implications in the zooplankton-fish relationship. Due to induced fear of predation, there can be a reduced reproduction rate of zooplankton species, and the effect of this non-lethal interconnection can be carried over to subsequent seasons or generations. In the current study, we tend to analyse the role of fish-induced fear in zooplankton with its carry-over effects (COEs) and a corresponding discrete delay (COE delay) in a phytoplankton-zooplankton-fish population model. We use Holling type IV and II functional responses to model the phytoplankton-zooplankton and zooplankton-fish interplay, respectively. In the well-posedness of the present biological system, firstly, we evaluate an invariant set in which the solutions of the model remain bounded. Then we prove its persistence under some ecologically well-behaved conditions. Next, we establish the conditions under which the different feasible equilibrium points exist; the existence of various interior equilibria is also set up here. To study the system's dynamical behavior, local and global stability analyses for the equilibria mentioned above are also discussed. Further, the theoretical conditions for Hopf and transcritical bifurcations in non-delayed and delayed models are determined. Impacts of non-lethal parameters, fear, and its carry-over effects, on the population densities are studied analytically and supported numerically. For intermediate values of COEs parameter, we notice that the system behaves chaotically, and decreasing (or increasing) it to low (or high) values, solution converges to interior equilibrium point through period-halving. Calculation of the largest Lyapunov exponent and drawing of Poincaré map validate the chaotic nature of the system. The chaos for medium values of COEs parameter can also be controlled by decreasing the fear parameter. Next, we numerically validate the theoretical result for transcritical bifurcation. We also note that our system shows the phenomenon of enrichment of paradox, and the attribute of multistability. In the delayed model, we observe that increasing delay can eliminate chaotic oscillations through amplitude death phenomenon. We draw various types of graphs and diagrams to assist our results. Thus we can say that the present study has various interesting characters related to non-linear models and can help biologists to study the plankton-fish models in a more detailed and pragmatic manner.

Numerical Simulations on Nonlinear Quantum Graphs with the GraFiDi Library

Nonlinear quantum graphs are metric graphs equipped with a nonlinear Schrödinger equation. Whereas in the last ten years they have known considerable developments on the theoretical side, their study from the numerical point of view remains in its early stages. The goal of this paper is to present the Grafidi library, a Python library which has been developed with the numerical simulation of nonlinear Schrödinger equations on graphs in mind. We will show how, with the help of the Grafidi library, one can implement the popular normalized gradient flow and nonlinear conjugate gradient flow methods to compute ground states of a nonlinear quantum graph. We will also simulate the dynamics of the nonlinear Schrödinger equation with a Crank-Nicolson relaxation scheme and a Strang splitting scheme. Finally, in a series of numerical experiments on various types of graphs, we will compare the outcome of our numerical calculations for ground states with the existing theoretical results, thereby illustrating the versatility and efficiency of our implementations in the framework of the Grafidi library.

Making graphs compact by lossless contraction

This paper proposes a scheme to reduce big graphs to small graphs. It contracts obsolete parts and regular structures into supernodes. The supernodes carry a synopsis S_mathcal {Q} for each query class mathcal {Q} in use, to abstract key features of the contracted parts for answering queries of mathcal {Q}. Moreover, for various types of graphs, we identify regular structures to contract. The contraction scheme provides a compact graph representation and prioritizes up-to-date data. Better still, it is generic and lossless. We show that the same contracted graph is able to support multiple query classes at the same time, no matter whether their queries are label based or not, local or non-local. Moreover, existing algorithms for these queries can be readily adapted to compute exact answers by using the synopses when possible and decontracting the supernodes only when necessary. As a proof of concept, we show how to adapt existing algorithms for subgraph isomorphism, triangle counting, shortest distance, connected component and clique decision to contracted graphs. We also provide a bounded incremental contraction algorithm in response to updates, such that its cost is determined by the size of areas affected by the updates alone, not by the entire graphs. We experimentally verify that on average, the contraction scheme reduces graphs by 71.9% and improves the evaluation of these queries by 1.69, 1.44, 1.47, 2.24 and 1.37 times, respectively.

An Expanded Concept and Graph of Zadeh’s Algebraic Operations for 3‐Dimensional Generalized Triangular Fuzzy Sets

Algebraic operators for fuzzy sets have been studied in 2‐dimensional space. We define the algebraic operators between two 3‐dimensional fuzzy sets and calculate the operators for the generalized 3‐dimensional triangular fuzzy sets. To facilitate an understanding of the results in 3D, the values of membership function are displayed in graphs using color density. We give various types of graphs of α‐cuts.

Graph with respect to superfluous elements in a lattice

We consider superfluous elements in a bounded lattice with 0 and 1, and introduce various types of graphs associated with these elements. The notions such as superfluous element graph (S(L)), join intersection graph (JI(L)) in a lattice, and in a distributive lattice, superfluous intersection graph (SI(L)) are defined. Dual atoms play an important role to find connections between the lattice-theoretic properties and those of corresponding graph-theoretic properties. Consequently, we derive some important equivalent conditions of graphs involving the cardinality of dual atoms in a lattice. We provide necessary illustrations and investigate properties such as diameter, girth, and cut vertex of these graphs.

Seminormality, canonical modules, and regularity of cut polytopes

Motivated by a conjecture of Sturmfels and Sullivant we study normal cut polytopes . After a brief survey of known results for normal cut polytopes it is in particular observed that for simplicial and simple cut polytopes their cut algebras are normal and hence Cohen–Macaulay. Moreover, seminormality is considered. It is shown that the cut algebra of K 5 is not seminormal which implies again the known fact that it is not normal. For normal Gorenstein cut algebras and other cases of interest we determine their canonical modules. The Castelnuovo–Mumford regularity of a cut algebra is computed for various types of graphs and bounds for it are provided if normality is assumed. As an application we classify all graphs for which the cut algebra has regularity less than or equal to 4.

Improved Computational Approaches and Heuristics for Zero Forcing

Zero forcing is a graph coloring process based on the following color change rule: all vertices of a graph [Formula: see text] are initially colored either blue or white; in each timestep, a white vertex turns blue if it is the only white neighbor of some blue vertex. A zero forcing set of [Formula: see text] is a set of blue vertices such that all vertices eventually become blue after iteratively applying the color change rule. The zero forcing number [Formula: see text] is the cardinality of a minimum zero forcing set. In this paper, we propose novel exact algorithms for computing [Formula: see text] based on formulating the zero forcing problem as a two-stage Boolean satisfiability problem. We also propose several heuristics for zero forcing based on iteratively adding blue vertices which color a large part of the remaining white vertices. These heuristics are used to speed up the exact algorithms and can also be of independent interest in approximating [Formula: see text]. Computational results on various types of graphs show that, in many cases, our algorithms offer a significant improvement on the state-of-the-art algorithms for zero forcing. Summary of Contribution: This paper proposes novel algorithms and heuristics for an NP-hard graph coloring problem that has numerous applications. Our exact methods combine Boolean satisfiability modeling with a constraint generation framework commonly used in operations research. The paper also includes an analysis of the facets of the polytope associated with this problem and decomposition techniques which can reduce the size of the problem. Our computational approaches are implemented and tested on a wide variety of graphs and are compared with the state-of-the-art algorithms from the literature. We show that our proposed algorithms based on Boolean satisfiability, in conjunction with the heuristics and order-reduction techniques, yield a significant speedup in some cases.

Graphkit-learn: A Python library for graph kernels based on linear patterns

• Graph kernels bridge the gap between machine learning and data encoded as graphs. • Graph kernels based on linear patterns provide good performance. • A Python library for efficient computation of graph kernels is proposed. • Three strategies are provided to reduce computational complexity. • Experiments on synthesized and real-world datasets show the relevance of the library. This paper presents graphkit-learn , the first Python library for efficient computation of graph kernels based on linear patterns, able to address various types of graphs. Graph kernels based on linear patterns are thoroughly implemented, each with specific computing methods, as well as two well-known graph kernels based on non-linear patterns for comparative analysis. Since computational complexity is an Achilles’ heel of graph kernels, we provide several strategies to address this critical issue, including parallelization, the trie data structure, and the FCSP method that we extend to other kernels and edge comparison. All proposed strategies save orders of magnitudes of computing time and memory usage. Moreover, all the graph kernels can be simply computed with a single Python statement, thus are appealing to researchers and practitioners. For the convenience of use, an advanced model selection procedure is provided for both regression and classification problems. Experiments on synthesized datasets and 11 real-world benchmark datasets show the relevance of the proposed library.

High-Order Structure Exploration on Massive Graphs

Modeling and exploring high-order connectivity patterns, also called network motifs, are essential for understanding the fundamental structures that control and mediate the behavior of many complex systems. For example, in social networks, triangles have been proven to play the fundamental role in understanding social network communities; in online transaction networks, detecting directed looped transactions helps identify money laundering activities; in personally identifiable information networks, the star-shaped structures may correspond to a set of synthetic identities. Despite the ubiquity of such high-order structures, many existing graph clustering methods are either not designed for the high-order connectivity patterns, or suffer from the prohibitive computational cost when modeling high-order structures in the large-scale networks. This article generalizes the challenges in multiple dimensions. First ( Model ), we introduce the notion of high-order conductance, and define the high-order diffusion core, which is based on a high-order random walk induced by the user-specified high-order network structure. Second ( Algorithm ), we propose a novel high-order structure-preserving graph clustering framework named HOSGRAP , which partitions the graph into structure-rich clusters in polylogarithmic time with respect to the number of edges in the graph. Third ( Generalization ), we generalize our proposed algorithm to solve the real-world problems on various types of graphs, such as signed graphs, bipartite graphs, and multi-partite graphs. Experimental results on both synthetic and real graphs demonstrate the effectiveness and efficiency of the proposed algorithms.

Minimal Embedding Dimensions of Rectangle k-Visibility Graphs

Bar visibility graphs were adopted in the 1980s as a model to represent traces, e.g., on circuit boards and in VLSI chip designs. Two generalizations of bar visibility graphs, rectangle visibility graphs and bar $k$-visibility graphs, were subsequently introduced. Here, we combine bar $k$- and rectangle visibility graphs to form rectangle $k$-visibility graphs (R$k$VGs), and further generalize these to higher dimensions. A graph is a $d$-dimensional R$k$VG if and only if it can be represented with vertices as disjoint axis-aligned hyperrectangles in $d$-space, such that there is an axis-parallel line of sight between two hyperrectangles that intersects at most $k$ other hyperrectangles if and only if there is an edge between the two corresponding vertices. For any graph $G$ and a fixed $k$, we prove that given enough spatial dimensions, $G$ has a rectangle $k$-visibility representation, and thus we define the minimal embedding dimension (MED) with $k$-visibility of $G$ to be the smallest $d$ such that $G$ is a $d$-dimensional R$k$VG. We study the properties of MEDs and find upper bounds on the MEDs of various types of graphs. In particular, we find that the $k$-visibility MED of the complete graph on $m$ vertices $K_m$ is at most $\lceil{m/(2(k+1))}\rceil,$ of complete $r$-partite graphs is at most $r+1,$ and of the $m^{\rm th}$ hypercube graph $Q_m$ is at most $\lceil{2m/3}\rceil$ in general, and at most $\lfloor{\sqrt{m}\,}\rceil$ for $k=0,~ m \ne 2.$

A categorial approach to reaction systems: First steps

In the literature, one encounters the intensely studied classical set-based reaction systems and the more recently introduced generalization to graph-based reaction systems where the considered graphs are directed, simple, and edge-labeled. In this paper, we propose a categorical approach to reaction systems so that a wider spectrum of data structures becomes available on which reaction systems can be based including various types of graphs and of graph-like structures like unlabeled graphs, vertex-labeled graphs, bipartite graphs, and a variety of types of hypergraphs. But also algebraic structures like monoids fit into the framework.