Graph Theory Concepts Research Articles

Identification of Graph Thinking in Solving Mathematical Problems Naturally

This study attempts to describe characteristics of graph thinking in solving a mathematical problem. Three students at the 10th grade of senior-high schools were involved as the subject. The data was collected from the result of an optimization problem task (OPT), video recording, interviews, and field notes. The results showed two major characteristics of graph thinking were found in solving the problem. First, students used the concept of graph theory to create a problem modelling. They were able to represent the information given in the problem in the form of graph. Second, students also used the concept of graph theory to create a problem modelling and search algorithm. The problem modelling was created as the students interpreted the problem by making connection between the objects in the form of an adjacency matrix and connectivity. In devising a plan, the students referred to the problem modelling to develop search algorithms. However, the algorithms were not entirely efficient. Some of them required the students to initially describe all answer possibilities. The algorithms constructed by the students referred to sequential and conditional algorithms. This study argues that graph-thinking skill can be developed through a learning process which involves students in the solving of open-ended problem to stimulate ideas of problem solving. By developing graph thinking ability, students will be able to analyse and reason information, express mathematical ideas, and have flexibility in solving a problem. These skills are urgently needed in the 21st century where rapid and continuous changes occur.

THE USAGE OF GRAPH THEORY CONCEPTS FOR THE COMPLEX NETWORKS ANALYSIS

The article considers and investigates the usage of graph theory concepts for the complex networks analysis. Abstract from their physical nature, the topological properties of these networks are considered, which significantly determine the functioning of networks and are the subject of study of complex networks. Each node of the network can be connected with other nodes by a certain number of connections that may have a direction, or nodes can be connected with each other by symmetrical connections. Also in modern systems of analysis and visualization of networks such concepts as degrees of vertices, ranking, clustering, modularity, algorithms of laying of graphs, etc. are widely used. To calculate the parameters of the network as a whole use the number of nodes, the number of edges, the geodetic distance between nodes, the average distance from one node to another, density – the ratio of the number of edges in the network to the maximum possible number of edges for a given number of nodes; number of triads, diameter of the network (maximum geodetic distance). Structural network analysis includes: click detection (subgroups that are more interconnected than other click nodes); identification of network components; finding bridges (nodes, the removal of which breaks the network into incoherent parts); groups of equivalent nodes (which have the most similar communication profiles). One of the areas of complex networks analysis is their visualization, which allows to obtain important information about the structure and properties of the network without accurate calculations. Software tools for the complex networks analysis support the calculation of all the described parameters of the nodes, the network as a whole, provide its structural analysis and visualization, work with different data formats. For the complex networks analysis, the main visualization tools are described, in the vast majority, freely distributed, free programs.

A Heuristic for Strategic Valve Placement

An iterative heuristic is proposed to improve the distribution of isolation valves within an existing water distribution network to decrease the magnitude of service interruptions while using the minimum number of valves required to isolate any individual segment within the system. The method takes advantage of graph theory concepts to create a valve augmentation scheme that provides gradual upgrades using the minimum number of new isolation valves at each step. The developed algorithm seeks to provide a tradeoff between an increase in the number of isolation valves and a reduction in water shortages resulting from disconnected pipe segments. The approach is applied to an actual water distribution network with known existing valve locations. The results demonstrate the feasibility and utility of the procedure for multiple operational constraints (i.e., maximum number of valves per segment, or maximum allowable water shortage evaluated over all segments). By use of an incremental performance target set by the user, the algorithm can prove beneficial even for utilities with limited financial resources.

A Strategy to Strengthen and Enhance the Telecommunication Network in Sri Lanka by Using Concepts of Graph Theory and Linear Programming Models

The telecommunication sector in Sri Lanka is one of the most dynamic sectors, contributing significantly, both directly and indirectly, to investment, employment, productivity, and overall economic growth. According to the transformation of the telecommunication network of Sri Lanka, this study proposes a structure to increase the average upload and download speeds by using some concepts in pure Mathematics. The research design has aligned into two categories as interpretation and optimization. They established by using the concept of Graph Theory and linear programming models respective. We forecast the public switched telephone network (PSTN) model for the local network. Then we ranked carrier systems of Sri Lanka according to their services without violating privacy. Next, we created the minimum traffic flow, local model, based on the traffic flows. Then we proposed cable systems around Sri Lanka. After that, we calculated the required signal towers for the uniform coverage by considering the population density. Linear programming models had generated to optimize the traffic flows. We could obtain six hundred twenty-five linear programming models due to different coefficients in each case. So, we have separated those models into three sectors: low, moderate, and high. Three examples had discussed by considering them. The solutions had evaluated by using the PHP Simplex calculator.

Complexity and variability analyses of motor activity distinguish mood states in bipolar disorder.

Changes in motor activity are core symptoms of mood episodes in bipolar disorder. The manic state is characterized by increased variance, augmented complexity and irregular circadian rhythmicity when compared to healthy controls. No previous studies have compared mania to euthymia intra-individually in motor activity. The aim of this study was to characterize differences in motor activity when comparing manic patients to their euthymic selves. Motor activity was collected from 16 bipolar inpatients in mania and remission. 24-h recordings and 2-h time series in the morning and evening were analyzed for mean activity, variability and complexity. Lastly, the recordings were analyzed with the similarity graph algorithm and graph theory concepts such as edges, bridges, connected components and cliques. The similarity graph measures fluctuations in activity reasonably comparable to both variability and complexity measures. However, direct comparisons are difficult as most graph measures reveal variability in constricted time windows. Compared to sample entropy, the similarity graph is less sensitive to outliers. The little-understood estimate Bridges is possibly revealing underlying dynamics in the time series. When compared to euthymia, over the duration of approximately one circadian cycle, the manic state presented reduced variability, displayed by decreased standard deviation (p = 0.013) and augmented complexity shown by increased sample entropy (p = 0.025). During mania there were also fewer edges (p = 0.039) and more bridges (p = 0.026). Similar significant changes in variability and complexity were observed in the 2-h morning and evening sequences, mainly in the estimates of the similarity graph algorithm. Finally, augmented complexity was present in morning samples during mania, displayed by increased sample entropy (p = 0.015). In conclusion, the motor activity of mania is characterized by altered complexity and variability when compared within-subject to euthymia.

Design multivariate statistical process control procedure in the case of Ethio cement

PurposeThis study aims to design a multivariate control chart that improves the applicability of the traditional Hotelling T2 chart. This new type of multivariate control chart displays sufficient information about the states and relationships of the variables in the production process. It is used to make better quality control decisions during the production process.Design/methodology/approachMultivariate data are collected at an equal time interval and are represented by nodes of the graph. The edges connecting the nodes represent the sequence of operation. Each node is plotted on the control chart based on their Hotelling T2 statistical distance. The changing behavior of each pair of input and output nodes is studied by the neural network. A case study from the cement industry is conducted to validate the control chart.FindingsThe finding of this paper is that the points and lines in the classic Hotelling T2 chart are effectively substituted by nodes and edges of the graph respectively. Nodes and edges have dimension and color and represent several attributes. As a result, this control chart displays much more information than the traditional Hotelling T2 control chart. The pattern of the plot represents whether the process is normal or not. The effect of the sequence of operation is visible in the control chart. The frequency of the happening of nodes is recognized by the size of nodes. The decision to change the product feature is assisted by finding the shortest path between nodes. Moreover, consecutive nodes have different behaviors, and that behavior change is recognized by neural network.Originality/valueModifying the classical Hotelling T2 control chart by integrating with the concept of graph theory and neural network is new of its kind.

Inverse fuzzy multigraphs and planarity with application in decision-making

Recently, in Almallah et al. (New Math Nat Comput, to appear) and Borzooei et al. (New Math Nat Comput 16(2):397–418, 2020), we defined the concept of inverse fuzzy graph as a generalization of graph, which is able to answer some problems that graph theory and fuzzy graph theory can not explain. And as we know planarity is very important and applicable concept in graph theory and fuzzy graph theory, so our motivation in this paper is discussion the importance of planarity in inverse fuzzy graphs. Now, in this paper we define the notion of inverse fuzzy multigraph and the concept of planarity on it by using the concepts of intersecting value and inverse fuzzy planarity value. Then we introduce some related theorems which determining upper bounds and lower bounds for the inverse fuzzy planarity value. After that we define the strong (weak) planarity of an inverse fuzzy multigraph and investigate related results. Finally, we give an application of inverse fuzzy multigraphs to decision-making how to reduce the cost of travel tours.

Distributed Control and Optimization of DC Microgrids: A Port-Hamiltonian Approach

This article proposes a distributed secondary control scheme that drives a dc microgrid to an equilibrium point where the generators share optimal currents, and their voltages have a weighted average of nominal value. The scheme does not rely on the electric system topology nor its specifications; it guarantees plug-and-play design and functionality of the generators. First, the incremental model of the microgrid system with constant impedance, current, and power devices is shown to admit a port-Hamiltonian (pH) representation, and its passive output is determined. The economic dispatch problem is then solved by the Lagrange multipliers method; the Karush-Kuhn-Tucker conditions and weighted-average formation of voltages are then formulated as the control objectives. We propose a control scheme that is based on the Control by Interconnection design philosophy, where the consensus-based controller is viewed as a virtual pH system to be interconnected with the physical one. We prove the regional asymptotic stability of the closed-loop system using Lyapunov and LaSalle theorems. Equilibrium analysis is also conducted based on the concepts of graph theory and economic dispatch. Finally, the effectiveness of the presented scheme for different case studies is validated with a test microgrid system, simulated in both MATLAB/Simulink and OPAL-RT environments.

Component Based Self-Healing Approach for Fault-Tolerant Data Aggregation in WSN

Node faults in WSNs are often attributed to battery energy drain outs. While data aggregation is widely acknowledged as an energy conservation tool; the aggregator nodes are susceptible to battery energy drain out faults. When an aggregator/ intermediary node in a data aggregation tree fails, the child nodes of the faulty node get detached from the aggregation tree and form disconnected nodes, termed as Affected Nodes (ANs) and/or subtrees. To tolerate such faults, we use the concepts of component-based graph theory and consider the subtrees as components that need to be connected back to the root-component tree. Based on this, we develop two component based self-healing fault-tolerant algorithms SCR and SCR-DTRA, that exploit the inherent redundancy in WSNs to determine alternate paths from the affected components to the root node. Unlike most other approaches, the hallmark of the component-based algorithms is that they preserve precedence relations in the data aggregation hierarchy. The results of our simulations show that the two algorithms succeed in the recovery of more than 90% of ANs and subtrees affected due to faults. In terms of longevity, the two algorithms, last more than twice the number of rounds compared to other algorithms used for comparison.

ОСНОВНЫЕ АСПЕКТЫ МАРШРУТИЗАЦИИ В НАВИГАЦИОННЫХ СИСТЕМАХ НА ПРИМЕРЕ КОМПАНИИ «HERE TECHNOLOGIES»

This study is aimed at establishing the foundations of the scientific and theoretical base of naviga-tion mapping. Navigation mapping is one of the leading areas in cartography, which has been rapidly developing thanks to satellite and digital technologies. Navigation mapping activities resulted in navi-gation applications for motorists, for a wide range of consumers, various emergency services, fire de-partments, ambulance services. This wide range of navigation applications is used via smart phones, personal computers, walking and hand-held navigators. In addition to orientation on the terrain, the navigation system allows routing: to calculate the shortest or most profitable route, to guide you along the route. Road routing works in accordance with the algorithm based on previously created road graph. The route is built according to certain rules, taking into account the static attributes of the road network and dynamic geospatial factors. These aspects of routing are covered by the example of HERE Technologies. HERE Technologies is a large international navigation company for the creation, updating and use of navigation charts. The article also discusses the main static attributes of the navi-gation system, which are involved in building a route both online and offline; analyzed the main dy-namic factors of geospace, influencing the construction and adjustment of the route. The research uses the cartographic research method, the routing method and the basic concepts of graph theory.

Update article: applicability of artificial neural networks to integrate socio-technical drivers of buildings recovery following extreme wind events.

In a companion article, previously published in Royal Society Open Science, the authors used graph theory to evaluate artificial neural network models for potential social and building variables interactions contributing to building wind damage. The results promisingly highlighted the importance of social variables in modelling damage as opposed to the traditional approach of solely considering the physical characteristics of a building. Within this update article, the same methods are used to evaluate two different artificial neural networks for modelling building repair and/or rebuild (recovery) time. By contrast to the damage models, the recovery models (RMs) consider (A) primarily social variables and then (B) introduce structural variables. These two models are then evaluated using centrality and shortest path concepts of graph theory as well as validated against data from the 2011 Joplin tornado. The results of this analysis do not show the same distinctions as were found in the analysis of the damage models from the companion article. The overarching lack of discernible and consistent differences in the RMs suggests that social variables that drive damage are not necessarily contributors to recovery. The differences also serve to reinforce that machine learning methods are best used when the contributing variables are already well understood.

Strategy on Disaster Recovery Management based on Graph Theory Concepts

The COVID-19 pandemic has asserted major baseline facts from disaster anthropology during the last three decades. Resilience could be based on the solution to the question: "What is the maximum amount of destruction, if any, that the graph (a network) can sustain while ensuring that at least one of each technology type remains and that the remaining induced subgraph is properly colored?" The concept of a graph's Chromatic Core Subgraph is a solution to the stated problem. In this paper, the pandemic graphs and certain sequential graphs are developed. For these graphs, the Chromatic core subgraph is obtained. The results of the pandemic graphs' Chromatic core subgraph are used to develop a disaster recovery strategy for the COVID-19 pandemic.

Design of a Numerical Simulator for Finned-Tube Heat Exchangers with Arbitrary Circuitry

A numerical simulator that can handle a finned-tube heat exchanger with a complex refrigerant circuitry is presented. The model is based on a tube-by-tube approach in which each tube is considered as the control volume in the analysis of the mass, energy, and momentum conservation. A mathematical representation of the refrigerant circuitry is also developed using the concepts in graph theory. The tube–tube adjacency matrix provides a unique representation of the tube connectivity and can also be utilized for constraints handling. With the implementation of the constraints, nonphysical and infeasible circuitries are removed from the solution space, which is advantageous for an evolutionary search for an optimum solution. The numerical simulator is verified against two sets of experimental data, and the calculated heat duty and pressure drop are within ±6% and ±20%, respectively. Sample simulations of the performance of different refrigerants in three circuitry configurations, and circuitry optimization are conducted to investigate the reliability and effectiveness of the tube-tube adjacency matrix and simulation model in the search for an optimal solution.

Exploring the similarity between Han’s and non-Han’s Yuan poetry: Resistance distance metrics over character co-occurrence networks

Abstract The Yuan dynasty is the first non-Han unified dynasty in ancient China. It is subtle to tell the similarities and differences between the classic Chinese poems in the Yuan dynasty written by the writers of Han nationality and non-Han minority nationalities. By graph theory and statistics, this article establishes a novel method for quantifying the similarity between the Han’s and non-Han’s Yuan poetry. First, we formulate the character co-occurrence networks (CCONs) of the Han’s and non-Han’s Yuan poem databases collected from ‘Database of Ancient Chinese Classics’. The resistance distance, a graph theory concept, is introduced to describe the lexical relation between a pair of characters in the CCONs. Then, we propose the similarity index as the coefficient of variation for the ratios of resistance distances in the CCONs of Han’s and non-Han’s poem databases, which measures the consistency between the usages of characters in the Han’s and non-Han’s poems. The calculation results indicate that the Han’s and non-Han’s poems share a high percentage of common high-frequency characters and there do not exist significantly different lexical features in terms of the usage of these high-frequency characters. The obtained results offer new insights into the quantitative features of this long-debated issue.

Application of Mathematics in Computer Science

No one escape the learning of mathematics in one way or other, ranging from our kitchen to our journey from earth to Moon or Mars. Mathematics persists everywhere around us. It can be perceived in our garden or park from symmetry of leaves, flowers, fruits etc. and by so many examples of Geometry and symmetry can be seen in nature. God used mathematics in creation of the universe in one form or the other. Likewise, Mathematics is the queen of all sciences. Scientists and researchers can not perfectly accomplish their work without including mathematics. Mathematics is the foundation of Computer Science. If one is eager to learn any arena of Computer Science, first he/she has to imbibe a love of Mathematics that will be supportive for progressive learning of the said subject. Mathematics is friendly for analytical skills needed in Computer Science. Concepts of binary number system, Boolean algebra, Calculus, Discrete mathematics, linear algebra, number theory, and graph theory are the most applicable to the subject of computer science with the emergence of new concepts like machine learning, artificial intelligence, virtual reality and augmented reality.

Uncovering Effects of Hot Potatoes in Banking System: Arresting Die-Hard Issues

This study is aimed to conceptually engineer the effects of hot potatoes (bad loans) in the banking system. The study’s research design uses elementary concepts of graph theory comprising of mappings of systemic behavior of contemporary banking. Issues have been formally underpinned, described, and visualized through directed and linear graphs. Findings of the study show that it is an inherent feature of the banking system to generate hot potatoes and allow them to sit there. It creates heat in the system by way of latency, turgescence, propagation, and concurrent cyclicality. These predicaments of systemic crises of banking deteriorate the sustainability of the system. The pilot study (using classical t-Test) conducted as part of this research has validated the theoretical argument aforementioned. The study provides new insights into these die-hard issues. It helps refine the agenda of future research in banking.

Eulerian and Even-Face Graph Partial Duals

Eulerian and bipartite graph is a dual symmetric concept in Graph theory. It is well-known that a plane graph is Eulerian if and only if its geometric dual is bipartite. In this paper, we generalize the well-known result to embedded graphs and partial duals of cellularly embedded graphs, and characterize Eulerian and even-face graph partial duals of a cellularly embedded graph by means of half-edge orientations of its medial graph.

Analysis on sustainable manufacturing criteria of automotive industry

In this paper, criteria within the most significant domain of manufacturing in automotive industry in Malaysia is analysed and prioritized. This paper aims to propose a multi criteria decision making (MCDM) procedure using concept of fuzzy theory, graph theory and Laplacian matrix approach which is embedded into Graph Theory and Matrix approach (GTMA). The manufacturing domains are ranked using Laplacian energy while the criteria within the most significant domain is prioritized using method of graph drawing whereby the criteria’s is ranked based on the average distance among other criteria in Cartesian coordinate system. The result indicates that cost, quality and service in economic domain are among significant criteria’s that practitioners should give an attention in developing a quality sustainable manufacturing framework so that the industry could sustain for a longer period of time.

A Preliminary Study of Abnormal Centrality of Cortical Regions and Subsystems in Whole Brain Functional Connectivity of Autism Spectrum Disorder Boys.

The abnormal cortices of autism spectrum disorder (ASD) brains are uncertain. However, the pathological alterations of ASD brains are distributed throughout interconnected cortical systems. Functional connections (FCs) methodology identifies cooperation and separation characteristics of information process in macroscopic cortical activity patterns under the context of network neuroscience. Embracing the graph theory concepts, this paper introduces eigenvector centrality index (EC score) ground on the FCs, and further develops a new framework for researching the dysfunctional cortex of ASD in holism significance. The important process is to uncover noticeable regions and subsystems endowed with antagonistic stance in EC-scores of 26 ASD boys and 28 matched healthy controls (HCs). For whole brain regional EC scores of ASD boys, orbitofrontal superior medial cortex, insula R, posterior cingulate gyrus L, and cerebellum 9 L are endowed with different EC scores significantly. In the brain subsystems level, EC scores of DMN, prefrontal lobe, and cerebellum are aberrant in the ASD boys. Generally, the EC scores display widespread distribution of diseased regions in ASD brains. Meanwhile, the discovered regions and subsystems, such as MPFC, AMYG, INS, prefrontal lobe, and DMN, are engaged in social processing. Meanwhile, the CBCL externalizing problem scores are associated with EC scores.

A study on labelling of pythagorean neutrosophic fuzzy graphs

Pythagorean neutrosophic fuzzy set comprises elements with dependent membership (µ), non-membership (σ) and independent indeterminacy (β) functions with the flexibility 0 ≤ µ2 + β2 + σ2 ≤ 2. Pythagorean neutrosophic fuzzy graph is a new concept emerged by combining the concept of Pythagorean neutrosophic fuzzy set and fuzzy graph theory. In this paper, the authors present the labelling of Pythagorean neutrosophic fuzzy graphs and investigate their properties.