Graph Theory Concepts Research Articles

Network science and the human brain: Using graph theory to understand the brain and one of its hubs, the amygdala, in health and disease.

Over the past 15 years, the emerging field of network science has revealed the key features of brain networks, which include small-world topology, the presence of highly connected hubs, and hierarchical modularity. The value of network studies of the brain is underscored by the range of network alterations that have been identified in neurological and psychiatric disorders, including epilepsy, depression, Alzheimer's disease, schizophrenia, and many others. Here we briefly summarize the concepts of graph theory that are used to quantify network properties and describe common experimental approaches for analysis of brain networks of structural and functional connectivity. These range from tract tracing to functional magnetic resonance imaging, diffusion tensor imaging, electroencephalography, and magnetoencephalography. We then summarize the major findings from the application of graph theory to nervous systems ranging from Caenorhabditis elegans to more complex primate brains, including man. Focusing, then, on studies involving the amygdala, a brain region that has attracted intense interest as a center for emotional processing, fear, and motivation, we discuss the features of the amygdala in brain networks for fear conditioning and emotional perception. Finally, to highlight the utility of graph theory for studying dysfunction of the amygdala in mental illness, we review data with regard to changes in the hub properties of the amygdala in brain networks of patients with depression. We suggest that network studies of the human brain may serve to focus attention on regions and connections that act as principal drivers and controllers of brain function in health and disease.

The Properties of Topological Network Connectivity Measures and Their Application to U.S. Urban Freeway Networks

The mathematical concepts of graph theory were introduced into geography in the early 1960s, providing a means of conceptualizing transport networks as made up of nodes and links. Among the contributions were several measures of topological network connectivity known as α (alpha), β (beta), and γ (gamma). These ratio measures identified the level of connectivity present in a network. These measures attracted considerable attention before fading from view, although they have recently returned to prominence in several applications. Despite this renewed attention, the basic mathematical properties of these measures remain largely unexamined. This research investigates the properties of these three simple ratio measures and explores values and properties that might be typical in urban freeway networks. Although simple correlation reveals little relationship between connectivity and freeway structure, they can be used to identify several groups of freeway networks as well as identifying important details about the use and interpretation of these measures. It is likely that these simple measures will continue to be applied to new situations, and this article provides some guidelines for those applications.

Product of bipolar fuzzy graphs and their degree

The concepts of graph theory are applied in many areas of computer science including data mining, image segmentation, clustering, image capturing and networking. It is also known that lots of uncertainties occur in these areas. To handle the uncertainty that occurs in graph theory, fuzzy graph theory is successfully used in many problems. A bipolar fuzzy set is a generalization of the fuzzy set. In this paper, two new operations on bipolar fuzzy graphs, viz. normal product and tensor product, are defined. Also, the degrees of the vertices of the resultant graphs which are obtained from two given bipolar fuzzy graphs and using the operations Cartesian product, composition, tensor and normal product are determined.

Research and development of spectral clustering algorithms

Spectral clustering is a very competitive clustering method. It attracts more and more attention of academia in recent years and becomes a new hotspot in machine learning since it does not make any assumptions on the global structure of data and has excellent performance for the sample space of arbitrary shape. The idea of spectral clustering is based on spectral graph theory. It treats data clustering problem as a graph partitioning problem and we can get the best clustering results by constructing a suitable graph, and using the appropriate graph cut method. Spectral clustering is better than traditional clustering algorithms in many aspects and has been successfully applied in areas such as data analysis, speech and image processing. This paper first introduces the basic concepts of graph theory and traditional graph cut methods; then analyses typical spectral clustering algorithms and reviews the latest development of spectral clustering; finally, proposes several valuable research directions in light of the deficiencies of spectral clustering algorithms.

Graph Coloring Problems in Modern Computer Science

Graph coloring is one of the most important concepts in graph theory and is used in many real time applications in computer science. The main aim of this paper is to present the importance of graph coloring ideas in various areas of compute applications for researches that they can use graph coloring concepts for the research. Graph coloring used in various research areas of computer science such data mining, image segmentation, clustering, image capturing, networking etc. This papers mainly focused on important applications such as Guarding an Art Gallery, Physical layout segmentation, Round-Robin Sports Scheduling, Aircraft scheduling, Biprocessor tasks, Frequency assignment, Final Exam Timetabling as a Grouping Problem, Map coloring and GSM mobile phone networks, and Student Time Table. In this paper we review several variants of graph colouring, such as precolouring extension, list colouring, multicolouring, minimum sum colouring, and discuss their applications in scheduling. A very important graph parameter is the chromatic number. Presently, graph coloring plays an important role in several real-world applications and still engages exciting research.

Graph Coloring Problems in Modern Computer Science

Graph coloring is one of the most important concepts in graph theory and is used in many real time applications in computer science. The main aim of this paper is to present the importance of graph coloring ideas in various areas of compute applications for researches that they can use graph coloring concepts for the research. Graph coloring used in various research areas of computer science such data mining, image segmentation, clustering, image capturing, networking etc. This papers mainly focused on important applications such as Guarding an Art Gallery, Physical layout segmentation, Round-Robin Sports Scheduling, Aircraft scheduling, Biprocessor tasks, Frequency assignment, Final Exam Timetabling as a Grouping Problem, Map coloring and GSM mobile phone networks, and Student Time Table. In this paper we review several variants of graph colouring, such as precolouring extension, list colouring, multicolouring, minimum sum colouring, and discuss their applications in scheduling. A very important graph parameter is the chromatic number. Presently, graph coloring plays an important role in several real-world applications and still engages exciting research.

Perfect Matching Quad Layouts for Manifold Meshes

AbstractThis paper introduces a new approach to automatically generate pure quadrilateral patch layouts on manifold meshes. The algorithm is based on a careful construction of a singularity graph of a given input frame field or a given periodic global parameterization. A pure quadrilateral patch layout is then derived as a constrained minimum weight perfect matching of that graph. The resulting layout is optimal relative to a balance between coarseness and geometric feature alignment. We formulate the problem of finding pure quadrilateral patch layouts as a global optimization problem related to a well‐known concept in graph theory. The main advantage of the new method is its simplicity and its computation speed. Patch layouts generated by the present algorithm are high quality and are very competitive compared to current state of the art.

Network analysis for a network disorder: The emerging role of graph theory in the study of epilepsy.

Recent years have witnessed a paradigm shift in the study and conceptualization of epilepsy, which is increasingly understood as a network-level disorder. An emblematic case is temporal lobe epilepsy (TLE), the most common drug-resistant epilepsy that is electroclinically defined as a focal epilepsy and pathologically associated with hippocampal sclerosis. In this review, we will summarize histopathological, electrophysiological, and neuroimaging evidence supporting the concept that the substrate of TLE is not limited to the hippocampus alone, but rather is broadly distributed across multiple brain regions and interconnecting white matter pathways. We will introduce basic concepts of graph theory, a formalism to quantify topological properties of complex systems that has recently been widely applied to study networks derived from brain imaging and electrophysiology. We will discuss converging graph theoretical evidence indicating that networks in TLE show marked shifts in their overall topology, providing insight into the neurobiology of TLE as a network-level disorder. Our review will conclude by discussing methodological challenges and future clinical applications of this powerful analytical approach.

A Survey on some Applications of Graph Theory in Cryptography

Graph theory is rapidly moving into the main stream of research because of its applications in diverse fields such as biochemistry (genomics), coding theory, communication networks and their security etc. In particular researchers are exploring the concepts of graph theory that can be used in different areas of Cryptography. In this paper a review of the works carried out in the field of Cryptography which use the concepts of Graph Theory, is given. Some of the Cryptographic Algorithms based on general graph theory concepts, Extremal Graph Theory and Expander Graphs are analyzed.

Fusion Trust Relation and Rating Data Algorithm

A new algorithm FTRA has been proposed, which infuses users’ trust network and rating data. The sparse problem of rating data will significantly reduce the accuracy of collaborative filtering recommendation. In addition to the users’ ratings data on the Internet, other data sources which can be used in the process of recommend, and one of the more common is trust network data which describes the mutual relationship between users. To solve this problem, this paper will the data of trust network as an important supplement on the rating data, and bases on graph theory concepts or methods, the similarity method in the paper, and the Katz method which is used to calculate the similarity of link, proposes the FTRA algorithm which organic infuses this two data, and then better to solve the sparse problem of the rating data faced by collaborative filtering. The experimental results on the Epinions dataset show that the FTRA algorithm is superior to or significantly better than the comparison algorithms, which include the algorithms that only based on the rating data or the trust relationship, and the other algorithms infusing the two data sources.

Complete hierarchies of SIR models on arbitrary networks with exact and approximate moment closure

We first generalise ideas discussed by Kiss et al. (2015) to prove a theorem for generating exact closures (here expressing joint probabilities in terms of their constituent marginal probabilities) for susceptible-infectious-removed (SIR) dynamics on arbitrary graphs (networks). For Poisson transmission and removal processes, this enables us to obtain a systematic reduction in the number of differential equations needed for an exact ‘moment closure’ representation of the underlying stochastic model. We define ‘transmission blocks’ as a possible extension of the block concept in graph theory and show that the order at which the exact moment closure representation is curtailed is the size of the largest transmission block. More generally, approximate closures of the hierarchy of moment equations for these dynamics are typically defined for the first and second order yielding mean-field and pairwise models respectively. It is frequently implied that, in principle, closed models can be written down at arbitrary order if only we had the time and patience to do this. However, for epidemic dynamics on networks, these higher-order models have not been defined explicitly. Here we unambiguously define hierarchies of approximate closed models that can utilise subsystem states of any order, and show how well-known models are special cases of these hierarchies.

Using graph theory for automated electric circuit solving

Graph theory plays many important roles in modern physics and in many different contexts, spanning diverse topics such as the description of scale-free networks and the structure of the universe as a complex directed graph in causal set theory. Graph theory is also ideally suited to describe many concepts in computer science. Therefore it is increasingly important for physics students to master the basic concepts of graph theory. Here we describe a student project where we develop a computational approach to electric circuit solving which is based on graph theoretic concepts. This highly multidisciplinary approach combines abstract mathematics, linear algebra, the physics of circuits, and computer programming to reach the ambitious goal of implementing automated circuit solving.

Three-dimensional gravity inversion using graph theory to delineate the skeleton of homogeneous sources

We developed a new strategy, based on graph theory concepts, to invert gravity data using an ensemble of simple point masses. Our method consisted of a genetic algorithm with elitism to generate a set of possible solutions. Each estimate was associated to a graph to solve the minimum spanning tree (MST) problem. To produce unique and stable estimates, we restricted the position of the point masses by minimizing the statistical variance of the distances of an MST jointly with the data-misfit function during the iterations of the genetic algorithm. Hence, the 3D spatial distribution of the point masses identified the skeleton of homogeneous gravity sources. In addition, our method also gave an estimation of the anomalous mass of the source. So, together with the anomalous mass, the skeleton could aid other 3D methods with promising geometric a priori parameters. Several tests with different values of regularizing parameter were made to bespeak this new regularizing strategy. The inversion results applied to noise-corrupted synthetic gravity data revealed that, regardless of promising starting models, the estimated distribution of point masses and the anomalous mass offered valuable information about the homogeneous sources in the subsurface. Tests on real data from a portion of Quadrilátero Ferrífero, Minas Gerais state, Brazil, were performed for complementary analysis of the proposed inversion method.

A new look at the topographical global optimization method and its application to the phase stability analysis of mixtures

Optimization methods that use optimality conditions of first and/or second order are known to be efficient. Commonly, such iterative methods are developed and analyzed in the light of knowledge concerning the mathematical analysis in n-dimensional Euclidean spaces, whose nature is of local character. Consequently, these methods lead to iterative algorithms that perform only local searches. Thus, the application of such algorithms to the calculation of global minimizers of a non-linear function, especially non-convex and multimodal, depends strongly on the location of the starting points. The Topographical Global Optimization method is a clustering algorithm, which uses an ingenious approach based on elementary concepts of graph theory, in order to generate good starting points for local search methods, from points distributed uniformly in the interior of the feasible set. The purpose of this paper is two-fold. The first is a revisit to the Topographical Global Optimization method, where, for the first time, its foundations are formally described and its basic properties are mathematically proven. In this context, we propose a semi-empirical formula for computing the key parameter of this clustering algorithm, and, using a robust and efficient direction interior-point method, we extend the use of the Topographical Global Optimization method to problems with inequality constraints. The second objective is the application of this method to the phase stability analysis of mixtures, a difficult and important global optimization problem of the chemical engineering thermodynamics. Furthermore, in order to have an initial assessment of the power of this technique, first we solve 70 test problems, and then compare the performance of the method considered here with the MIDACO solver, a powerful software recently introduced in the field of global optimization.

Different Design Criteria for District Metered Areas in Water Distribution Networks

In this paper the influence of several factors to establish a suitable DMA design is considered using an adaptation of the methodology previously developed by Gomes [1]. This methodology is based on graph theory concepts (Floyd-Warshall algorithm) and some user-defined criteria to establish the number and size of DMAs and uses a Simulated Annealing algorithm to identify the most appropriate number and location of metering stations, boundary valves and network reinforcement/replacement needs along the project plan. Different design criteria are applied to a case study and the results obtained by the methodology are used to draw some conclusions.

Basic Concepts of Graph Theory as the Instruments of Mathematical Formalization of Woven Structures

The paper presents the prerequisites of involving of topological elements and graph theory as an instrument of mathematical formalization of woven structures and technology of textile fabrics. Present research is based on analysis and comparison of the main concepts and conditions of textile technology and graph theory.

Locating Leaks in Water Distribution Networks with Simulated Annealing and Graph Theory

This paper addresses the problem of locating leaks in water distribution networks. The approach presented here is based on steady-state modelling, supported by monitoring tank flow and pressure at strategic nodes. The selection of the pressure monitoring nodes is done applying graph theory concepts adapted to water distribution networks. Pressure monitoring data is then used to build an optimization problem: the objective function is the minimization of the differences between estimated and measured pressures at the monitoring points, the constraints are the common energy and mass conservation laws and the decision variables are the leak locations and flows. This optimization problem is solved by a simulated annealing algorithm presented in a previous work. The application of this approach in a set of case studies produced some encouraging results and this paper describes the complete procedure and draws some conclusions.

Distributed coloring algorithms for triangle-free graphs

Vertex coloring is a central concept in graph theory and an important symmetry-breaking primitive in distributed computing. Whereas degree-Δ graphs may require palettes of Δ+1 colors in the worst case, it is well known that the chromatic number of many natural graph classes can be much smaller. In this paper we give new distributed algorithms to find (Δ/k)-coloring in graphs of girth 4 (triangle-free graphs), girth 5, and trees. The parameter k can be at most (14−o(1))ln⁡Δ in triangle-free graphs and at most (1−o(1))ln⁡Δ in girth-5 graphs and trees, where o(1) is a function of Δ. Specifically, for Δ sufficiently large we can find such a coloring in O(k+log⁎⁡n) time. Moreover, for any Δ we can compute such colorings in roughly logarithmic time for triangle-free and girth-5 graphs, and in O(log⁡Δ+logΔ⁡log⁡n) time on trees. As a byproduct, our algorithm shows that the chromatic number of triangle-free graphs is at most (4+o(1))Δln⁡Δ, which improves on Jamall's recent bound of (67+o(1))Δln⁡Δ. Finally, we show that (Δ+1)-coloring for triangle-free graphs can be obtained in sublogarithmic time for any Δ.

Complete graphs in the Rubáiyát

The Rubáiyát of Omar Khayyám has fascinated readers for centuries, and it has been translated and interpreted many times. In this paper, we will describe a few basic graph theory concepts, and discuss how graph theory can be used to explore the connections between the various quatrains contained in Edward FitzGerald's several translations of the Rubáiyát. We will explain the process of searching for certain complete subgraphs of the full graph of the Rubáiyát, and will briefly discuss how these ideas may be relevant in other areas. These applications include analysing other collections of poetry, teaching certain types of incidence geometry and poetic forms for composing short collections of poetry.

A new AATP model with considering supply chain lead-times and resources and scheduling of the orders in flowshop production systems: A graph-theoretic view

The present study attempts to synchronize the scheduling problem with determining the advanced available-to-promise (AATP) in a flowshop system to enhance supplier profitability and service level. In the proposed model the AATP, scheduling and graph theory concept have been combined to find the optimum resource allocation and enable accurate estimations of machines scheduling, production costs and delivery dates. To find the near optimum solutions for the large size problems a genetic algorithm is developed, first the orders are ranked based on their scores which are estimated then the optimum cost is calculated by balancing profitability and constraints such as the availability of the machines or the available material in each workstation. Some computer simulated experiments are provided to evaluate the performance of the proposed algorithm.