Graph Theory Concepts Research Articles

Graph Theory Applications in Database Management

Graph theory, which is a branch of discrete mathematics, has emerged as a powerful tool in various domains, including database management. This abstract investigates the ways in which ideas and methods from graph theory which can be applied to database systems, offering a thorough synopsis of their benefits. Complex interactions within data can be well-modeled by using the basic concepts of graph theory, such as nodes, edges, and relationships. Because of its capacity to represent and query complex relationships, graph databases have become more and more popular in the field of database administration. Graph databases are well-suited for situations such as social networks, recommendation systems, and interconnected data domains because they are excellent at representing and traversing relationships, in contrast to standard relational databases, which are excellent at managing structured data. The abstract delves into the key graph-based data models, such as property graphs, RDF (Resource Description Framework), explaining how they facilitate the representation of diverse relationships. Furthermore, it explores the efficient storage and retrieval mechanisms that leverage graph traversal algorithms to extract valuable insights from interconnected datasets. The document highlights specific use cases where graph theory contributes to database management, including fraud detection, social network analysis, and recommendation systems. Additionally, it discusses the challenges associated with integrating graph databases into existing infrastructures and proposes solutions to address scalability and performance concerns. The abstract also touches upon the advancements in graph database query languages (Cypher) and SPARQL, showcasing their expressive power in querying complex relationships. The inclusion of graph-based indexing and optimization techniques demonstrates how database systems can efficiently handle queries involving large-scale graph data. As graph databases continue to evolve, this abstract concludes by outlining potential future directions in the intersection of graph theory and database management. It emphasizes the importance of ongoing research in developing scalable and efficient solutions for managing interconnected data, ultimately paving the way for more sophisticated and context-aware database systems relationships. Furthermore, it explores the efficient storage and retrieval mechanisms that leverage graph traversal algorithms to extract valuable insights from interconnected datasets.

A branch-and-cut approach to solve the Fault Diagnosis Problem with Lazy Spread and imperfect system information

This paper presents a new approach to solving the Fault Diagnosis Problem with Lazy Spread (FDPL) that arises in many fault-tolerant real-world systems with few opportunities for maintenance during their operations and significant failure interactions between the subsystems/components. As opposed to cascading faults that spread to most of the system almost instantaneously, FDPL considers fault-resistant systems where the spread of failures is relatively slow (lazy), i.e., only a small fraction of the components are faulty at the time of inspection, and accurate diagnosis of the faulty components is of critical importance to restore system performance and stop further damage. Introducing an extra level of difficulty, FDPL needs to be solved under imperfect (missing and wrong) system information in most real-world settings. To address this challenging prediction problem, we use graph theory concepts to develop an integer programming formulation and devise an efficient branch-and-cut algorithm for its solution. Extensive numerical experiments on realistic problem instances attest to the superior performance of our approach in terms of both computational efficiency and prediction accuracy compared to the state-of-the-art in the literature.

On the metric-based resolving parameter of the line graph of certain structures

Let G be a graph and R = {r1, r2, …, rk} be an ordered subset of vertices of G, if every two vertices of G have different representation r (v|R) = (d (v, r1) , d (v, r2) , …, d (v, rk)) with respect to R, then R is said to be a metric-based resolving parameter or resolving set of G and its minimum cardinality is called the metric dimension of graph G . Metric dimension is considered as an important applied concept of graph theory especially in the localization of a network and also in the chemical graph theoretical study of molecular compounds. Therefore, it is hot topic to study for different families of graphs as well. Convex polytopes play an important role both in various branches of mathematics and in applied areas, most notably in linear programming. In this paper, we determine the metric-based resolving parameter of line graph of a convex polytope Sn, and conclude that it has constant metric dimension but vary with the parity of n . This article presents a measurement of the line graph of a convex polytope, denoted as ( S n ) . The subsequent section provides the metric dimension of the resulting graph. There are two scenarios pertaining to the metric dimension of a selected graph with respect to the metric dimension. The metric dimension of even cycle-based convex polytopes is three, whereas for other values, the metric dimension is four.

Detection and Validation of Macro-Activities in Human Inertial Signals Using Graph Link Prediction.

With the continuous development of new wearable devices, sensor-based human activity recognition is enjoying enormous popularity in research and industry. The signals from inertial sensors allow for the detection, classification, and analysis of human activities such as jogging, cycling, or swimming. However, human activity recognition is often limited to basic activities that occur in short, predetermined periods of time (sliding windows). Complex macro-activities, such as multi-step sports exercises or multi-step cooking recipes, are still only considered to a limited extent, while some works have investigated the classification of macro-activities, the automated understanding of how the underlying micro-activities interact remains an open challenge. This study addresses this gap through the application of graph link prediction, a well-known concept in graph theory and graph neural networks (GNNs). To this end, the presented approach transforms micro-activity sequences into micro-activity graphs that are then processed with a GNN. The evaluation on two derived real-world data sets shows that graph link prediction enables the accurate identification of interactions between micro-activities and the precise validation of composite macro-activities based on learned graph embeddings. Furthermore, this work shows that GNNs can benefit from positional encodings in sequence recognition tasks.

Recognizing LBFS trees of bipartite graphs

The graph searches Breadth First Search (BFS) and Depth First Search (DFS) and the spanning trees constructed by them are some of the most basic concepts in algorithmic graph theory. BFS trees are first-in trees, i.e., every vertex is connected to its first visited neighbor. DFS trees are last-in trees, i.e., every vertex is connected to the last visited neighbor before it. The problem whether a given spanning tree can be the first-in tree or last-in tree of a graph search ordering was introduced in the 1980s and has been studied for several graph searches and graph classes. Here, we consider the problem of deciding whether a given spanning tree of a bipartite graph can be a first-in tree or a last-in tree of the Lexicographic Breadth First Search (LBFS), a special variant of BFS that is commonly used in graph algorithms. We show that the recognition of both first-in trees and last-in trees of LBFS is NP-hard even if the start vertex of the search ordering is fixed and the height of the tree is four. We prove that the bound on the height is tight (unless P=NP) by showing that for all spanning trees of bipartite graphs with height smaller than four we can solve both search tree recognition problems of LBFS in polynomial time. Finally, we give a linear-time algorithm that solves both problems for chordal bipartite graphs and fixed start vertices.

The development of RBL-stem learning materials to improve students’ combinatorial thinking skills in solving b-coloring problems for companion farming

Research Based Learning can be packaged with STEM approach. In a RBL, students will find a problem that requires a solution, problem solving skills are very important in learning. One of these mathematical thinking skills is combinatorial thinking skills. b-coloring is one of the concepts of graph theory. b-Coloring is widely applied in various fields, one of which we can apply to companion farming precision agriculture The development of RBL-STEM learning tools to improve students' combinatorial thinking skills in this research meet the criteria of valid, practical, and effective. The validity scores was 3.87. The observation results of learning implementation were 3.83 with a percentage of 95.75%, and student responses were 96.5% positive so that they met practical criteria. Based on the test results, researchers found that 97.5% of students were complete so that they met the effective criteria. Quantitative analysis involves processing pretest and posttest data, where normality test and paired sample t-test are conducted. Based on the normality test, it can be concluded that the pretest and posttest scores have a normal distribution, because the p-value is greater than 0.05, namely 0.053 and 0.059. Furthermore, a paired sample t-test was conducted which showed a p-value of 0.000. These results show that there is a significant increase in students' combinatorial thinking skills after participating in the learning.

Equivalence of lattice operators and graph matrices

Abstract We explore the relationship between lattice field theory and graph theory, placing special emphasis on the interplay between Dirac and scalar lattice operators and matrices within the realm of spectral graph theory. Beyond delving into fundamental concepts of spectral graph theory, such as adjacency and Laplacian matrices, we introduce a novel matrix called an “antisymmetrized adjacency matrix”, specifically tailored for cycle digraphs (T1 lattice) and simple directed paths (B1 lattice). The nontrivial relationship between graph theory matrices and lattice operators shows that the graph Laplacian matrix mirrors the lattice scalar operator and the Wilson term in lattice fermions, while the antisymmetrized adjacency matrix, along with its extensions to higher dimensions, is equivalent to naive lattice Dirac operators. Building upon these connections, we provide rigorous proofs for two key assertions: (i) The count of zero-modes in a free lattice scalar operator coincides with the zeroth Betti number of the underlying graph (lattice). (ii) The maximum count of Dirac zero-modes in a free lattice fermion operator is equivalent to the cumulative sum of all Betti numbers when the D-dimensional graph results from a Cartesian product of cycle digraphs (T1 lattice) and simple directed paths (B1 lattice).

Game Theory-Based Incentive Design for Mitigating Malicious Behavior in Blockchain Networks

This paper presents an innovative incentive model that utilizes graph and game theories to address the issue of node incentives in decentralized blockchain networks such as EVM blockchains. The lack of incentives for nodes within EVM networks gives rise to potential weaknesses that might be used for various purposes, such as broadcasting fake transactions or withholding blocks. This affects the overall trust and integrity of the network. To address this issue, the current study offers a network model that incorporates the concepts of graph theory and utilizes a matrix representation for reward and trust optimization. Furthermore, this study presents a game-theoretic framework that encourages cooperative conduct and discourages malicious actions, ultimately producing a state of equilibrium according to the Nash equilibrium. The simulations validated the model’s efficacy in addressing fraudulent transactions and emphasized its scalability, security, and fairness benefits. This study makes a valuable contribution to the field of blockchain technology by presenting an incentive model that effectively encourages the development of secure and trusted decentralized systems.

Research on Opinion Dynamics on Social Networks

Opinion dynamics, which focuses on the formation, propagation, and evolution of people’s opinions in social networks, has recently been widely studied. This paper outlines the history of opinion dynamics research, current methods, and future directions. Specifically, some basic theoretical concepts of graph theory are initially introduced. Then, the key research findings of opinion dynamics, such as Voter’s and Degroot’s models, which are discrete and continuous models, are reviewed. Finally, the future direction of this field is discussed.

Enumeration of dissociation sets in grid graphs

<abstract><p>A dissociation set of a graph $ G $ refers to a set of vertices inducing a subgraph with maximum degree at most 1 and serves as a generalization of two fundamental concepts in graph theory: Independent sets and induced matchings. The enumeration of specific substructures in grid graphs has been a captivating area of research in graph theory. Over the past few decades, the enumeration problems related to various structures in grid graphs such as Hamiltonian cycles, Hamiltonian paths, independent sets, maximal independent sets, and dominating sets have been deeply studied. In this paper, we enumerated dissociation sets in grid graphs using the state matrix recursion algorithm.</p></abstract>

Lucky edge geometric mean labeling of graphs and applications in electrical networks

A Lucky Edge Geometric Mean Labeling (LEGML) is a function μ that assigns the integers to the vertices of graph G such that for every edge as ‘{E}’, uv∈E(G) there exists a function μ*:E(G) → N is defined by μ* (uv) = √μ(u)μ(v) (or) √μ(u)μ(v) with the state that μ* (uv) ≠ μ* (vw) whenever {u} and {vw} have a common vertex as {v}. A minor number ‘k→ E = {1, 2, 3 … k} is the LEGML of G and is signified by h¢GM(G). A result that accepts LEGML is the Lucky Edge Geometric Mean Graph (LEGMG). The idea of the Graph Theory (GT) concept is fused into the Electric Circuit (EC), and the related complicated problems are transformed into graph model representation problems; as a result, the valuation method is simplified and optimized. Network analysis means finding the current or voltage in each branch. One application of GT is the representation of EC. An ‘E’ in a diagram can signify anything in an EC. Each part of the diagram can represent a port or a terminal in the EC. In this article, we have to investigate the Middle Graph (MG) of the star, the Total Graph (TG) of the star, and the central graph (CG) of the star, which accepts LEGML and LEGMG. This method also computes the power consumed by all the resistors of the EC by LEGML.

Inbound traffic capture link-design problem independent of assumptions on users’ route choices

In some traffic management situations, a cordon (a set of points at which traffic flows into a given area) is set in a road network to establish a reference for the location of equipment to implement traffic measurements and controls (e.g., traffic volume surveys and congestion charging). However, few studies have focused on the optimum location of a cordon. We devise a problem denoted the inbound traffic capture link-design problem to select the optimum combination of links for inclusion in a cordon. We regard this combination as the minimum number of links that can capture traffic on all routes, under the condition that there is a path between nodes inside the cordon that is not captured. We formulate this model by employing the graph theory concept of the minimum cut, and use the concept of a Steiner tree with auxiliary network flows to express the constraint of ensuring that there is an uncaptured path inside a cordon. After a basic formulation, to obtain an identical cordon, we devise two subsidiary schemes. In addition, we perform a linear relaxation of our method to reduce its computational cost. The results of computational experiments confirm that our model selects the optimal cordon location formed by a combination of capturing links and also outputs an identical cordon as a boundary line of an area. As the model is computationally feasible, even when applied on a large network, we believe it will have a wide range of practical applications.

Students’ mathematical thinking of the tree concept: an integration of APOS with Tall’s three worlds of mathematics

ABSTRACT In this study, we investigate students’ mathematical thinking of the tree concept, a fundamental concept in graph theory, using a combination of the APOS theory and Tall’s three worlds of mathematics. The qualitative research describes here is a phenomenological study drawing on task-based interviews with 18 undergraduate mathematics students. The findings indicate that students preferred to think in the embodied world when solving tasks related to the tree concept. This type of thinking, despite its great value, should not prevent students from thinking in the symbolic and formal worlds. Furthermore, the findings indicate that almost half of the participants had difficulties with the acyclicity and connectivity concepts, which are necessary prior knowledge to help students develop Process and Object for the tree concept. The results also suggest that more opportunities could be provided for students to engage with tasks that related to Process and Object in the three mathematics worlds.

A Java Application for Teaching Graphs in Undergraduate Courses

Graph theory is a common topic that is introduced as part of the curricula of computing courses such as Computer Science, Computer Engineering, Data Science, Information Technology and Software Engineering. Understanding graphs is fundamental for solving many real-world problems, such as network routing, social network analysis, and circuit design; however, many students struggle to grasp the concepts of graph theory, as they often have difficulties in visualising and manipulating graphs. To overcome these difficulties, educational software can be used to aid in the teaching and learning of graph theory. This work focuses on the development of a Java system for graph visualisation and computation, called MaGraDa (Graphs for Discrete Mathematics), that can help both students and teachers of undergraduate or high school courses that include concepts and algorithms related to graphs. A survey on the use of this tool was conducted to explore the satisfaction level of students on a Discrete Mathematics course taken as part of a Computer Engineering degree at the University of Alicante (Spain). An analysis of the results showed that this educational software had the potential to enhance students’ understanding of graph theory and could enable them to apply these concepts to solve practical problems in the field of computer science. In addition, it was shown to facilitate self-learning and to have a significant impact on their academic performance.

Evaluating the Vulnerability of the Sydney Train Network by Comparing Access-based and Network Centrality Metrics

Operational incidents are a significant cause of unreliability on rail transit networks. These incidents cause major delays in services, impact passenger travel time, and have knock-on effects that interrupt other public transport services. Consequently, the vulnerability of the rail transit network is a crucial concern for managers and operators. This paper employs network vulnerability analysis to characterize individual critical stations in a railway network. The concepts of graph theory and person-weighted access are implemented to identify the critical nodes in the Sydney train and metro network, and the results are compared. In the first method, weighted and unweighted centrality measures are computed to find the most critical station. In particular, eigenvector centrality is used to identify the critical nodes by scoring all nodes in the network using the first eigenvector of the graph adjacency matrix. In the second approach, stations are ranked by the reduction of access before and after an incident. Finding of this study may have implications not only for the train operators and managers but also for the transit network planners to enhance the resilience of the public transport network.

Modeling dynamic social networks using concept of neighborhood theory

Dynamic social network analysis basically deals with the study of how the nodes and edges and associations among them within the network alter with time, thereby forming a special category of social network. Geometrical analysis has been done on various occasions, but there is a difference in the approximate distances of nodes. Snapshots for social networks are taken at each time slot and then are bound for these studies. The paper will discuss an efficient way of modeling dynamic social networks with the concept of neighborhood theory of cellular automata. So far, no model that uses the concept of neighborhood has been proposed to the best of our knowledge and the literature survey. Besides cellular automata that has been important tool in various applications has remained unexplored in the area of modelling. To this extent the paper, is the 1st attempt in modelling the social network that is evolving in nature. A link prediction algorithm based on some basic graph theory concepts has also been additionally proposed for the emergence of new nodes within the network. Theoretical and programming simulations have been explained in support to the model. Finally, the paper will discuss the model with a real-life scenario.

FSMx-Ultra: Finite State Machine Extraction From Gate-Level Netlist for Security Assessment

Numerous security vulnerability assessment techniques urge precise and fast finite state machines (FSMs) extraction from the design under evaluation. Sequential logic locking, watermark insertion, fault-injection assessment of a System-on-a-Chip (SoC) control flow, information leakage assessment, and reverse engineering at gate-level abstraction, to name a few, require precise FSM extraction from the synthesized netlist of the design. Unfortunately, no reliable solutions are currently available for fast and accurate extraction of FSMs from the highly unstructured gate-level netlist for effective security evaluation. The major challenge in developing such a solution is precise recognition of FSM state flip-flops in a netlist having a massive collection of flip-flops. In this paper, we propose FSMx-Ultra, a framework for extracting FSMs from extremely unstructured gate-level netlists. FSMx-Ultra utilizes state-of-the-art graph theory concepts and algorithms to distinguish FSM state registers from other registers and then constructs gate-level state transition graphs (STGs) for each identified FSM state register using automatic test pattern generation (ATPG) techniques. The results of our experiments on 14 open-source benchmark designs illustrate that FSMx-Ultra can recover all FSMs quickly and precisely from synthesized gate-level netlists of diverse complexity and size utilizing various state encoding schemes.

Data-driven structural analysis of small cell lung cancer transcription factor network suggests potential subtype regulators and transition pathways

Small cell lung cancer (SCLC) is an aggressive disease and challenging to treat due to its mixture of transcriptional subtypes and subtype transitions. Transcription factor (TF) networks have been the focus of studies to identify SCLC subtype regulators via systems approaches. Yet, their structures, which can provide clues on subtype drivers and transitions, are barely investigated. Here, we analyze the structure of an SCLC TF network by using graph theory concepts and identify its structurally important components responsible for complex signal processing, called hubs. We show that the hubs of the network are regulators of different SCLC subtypes by analyzing first the unbiased network structure and then integrating RNA-seq data as weights assigned to each interaction. Data-driven analysis emphasizes MYC as a hub, consistent with recent reports. Furthermore, we hypothesize that the pathways connecting functionally distinct hubs may control subtype transitions and test this hypothesis via network simulations on a candidate pathway and observe subtype transition. Overall, structural analyses of complex networks can identify their functionally important components and pathways driving the network dynamics. Such analyses can be an initial step for generating hypotheses and can guide the discovery of target pathways whose perturbation may change the network dynamics phenotypically.

Enhanced connectivity and mobility in liquid water: Implications for the high density liquid structure and its onset

In this work, we investigate simulated liquid water at ambient pressure in both stable and metastable supercooled conditions by means of a new order parameter we recently proposed, namely the node total communicability (NTC), based on graph theory concepts. We show that this order parameter is able to identify the two liquid states differing in density, the LDL- and HDL-like states, in simulation conditions at which both states coexist. We also show that NTC is able to capture both the structural and dynamic differences between the two states, being correlated with both the local density and the mobility of water molecules within the network. In addition, we further investigate the high connectivity patches we previously identified as characteristic of the HDL-like state. We show that these extended patches are composed of molecules with an increased local density and mobility, packed in a highly connected network. The formation of these highly connected networks is characterized by a fast dynamics, with mobile molecules entering and exiting the patches. Interestingly, we observe small highly connected patches also at low temperatures, where the prevailing state is LDL-like. We show that the small-to-large patches transition is related to the Widom line crossing and we suggest that the small highly connected patches at low temperatures might function as initial sites for the formation of extended HDL-like regions characteristic of the highest temperatures.

Minimal cut of strongly connected directed random graphs studied in virtual machine deployment

Abstract This paper first presents the basic concepts of random graph theory, including undirected, directed, and generalized random graphs. A random homomorphic number mapping is introduced to solve the cut distance problem of directed random graphs, and a minimum cut distance is proposed. Next, a virtual machine deployment strategy (VMDS) based on the minimum cut algorithm is proposed to construct VM clusters based on similarity and to cut VM clusters based on the minimum cut of directed random graphs. According to the results, the MAE (1.3875) and MSE (2.7783) of both the conventional AR algorithm and ES algorithm are the lowest. Compared to the MAE (1.7381) and MSE (3.7840) of the VMDS algorithm, both significantly decrease. The research results presented in this paper are useful for applying virtual machine deployment techniques in cloud computing.