Graph Theory Concepts Research Articles

An Approach of Graph Theory on Cryptography

Abstract: In this paper, we discuss about the connection between graph theory and cryptography. We use the spanning tree concept of graph theory to encryption the message

A joint modeling and exploratory framework for intra-firm collaboration within construction and mining equipment industry

PurposeIn this research, collaboration attributes related to the firm's intrinsic and extrinsic facets at pertinent levels (i.e. enterprise, strategic, operational, and tactical levels) for construction equipment OEMs (original equipment manufacturers) operating in India have been quantified and modeled.Design/methodology/approachFor modeling the intra-firm collaboration at respective organizational levels, relevant attributes have been populated employing literature review followed by subsequent validation from pertinent focus groups. The focus groups comprising professionals working in the construction and mining equipment industry in India aided us in estimating the extent of interdependencies and influences within/amongst collaboration attributes. The collaboration attributes and respective interdependencies/influences are modeled employing the concept of graph theory wherein the individual attributes are represented using vertices and influences/interdependencies are represented using edges. The collaboration indices resulting from the variable permanent matrix have been derived as well.FindingsScenario and subsequent sensitivity analysis are performed. This research discusses the significance and aspects related to various collaborative attributes and the interrelations amongst them. Further, the research also evolves quantitative measures of collaboration indices at enterprise, strategic, tactical and operational levels by employing a graph-theoretic approach (GTA). The authors have also extricated and discussed a number of meaningful implications from both the perspectives of interorganizational relationships (IORs) and the normative theory of organizations using a cross-case analysis of five firms having operations in India.Originality/valueThe research would aid organizations (particularly those belonging to the construction equipment sector) measure the efficacy of collaboration in respective value-chains at strategic, tactical and operational levels. From the theoretical perspective, the integration of the IORs and normative theory of organizations enables looking at the intra-firm collaboration problem from a multi-dimensional standpoint involving activities, performance measures, action initiation, communication, shades of top management, level of activity, etc.

Adaptive sliding mode controller based consensus protocol for swarm of UAVs

PurposeThe requirement of robust cooperative control is essential to achieve consensus between unmanned aerial vehicles (UAVs) operating in swarm formation. Often the performance of these swarm formations is affected by wind gust disturbances. This study proposes an effective robust consensus protocol, which will ensure the UAVs in swam formation to collectively meet the desired objective in real-time scenario.Design/methodology/approachIn this work, the swarm of UAVs are modeled as multiagent systems by using the concepts of algebraic graph theory. To address the challenges of a complex and dynamic environment, an adaptive sliding mode control (SMC)-based consensus protocol is proposed. The closed loop stability analysis is established through Lyapunov theory.FindingsThe efficacy of the discussed robust consensus controller is analyzed through numerical simulations. Further, the quantitative analysis using Monte-Carlo simulations validates performance of the proposed robust consensus protocol. The presented consensus protocol can be easily implementable as robust flight controller for swarm of UAVs. Also, as the consensus theory is based on the algebraic graph theory, the proposed design is scalable for a large number of UAVs in swarm formation.Originality/valueThe proposed adaptive SMC achieves robust consensus of longitudinal dynamics states between all the UAVs by mitigating the effects of wind gust disturbances. Also, the adaptive SMC offers chattering-free control efforts.

Bridge Network Seismic Risk Assessment Using ShakeMap/HAZUS with Dynamic Traffic Modeling

Bridge infrastructures are critical nodes in a transportation network. In earthquake-prone areas, seismic performance assessment of infrastructure is essential to identify, retrofit, reconstruct, or, if necessary, demolish the infrastructure systems based on optimal decision-making processes. As one of the crucial components of the transportation network, any bridge failure would impede the post-earthquake rescue operation. Not only the failure of such high-risk critical components during an extreme event can lead to significant direct damages, but it also affects the transportation road network. The consequences of these secondary effects can easily lead to congestion and long queues if the performance of the transportation system before or after an event was not analyzed. These indirect losses can be more prominent compared to the actual damage to bridges. This paper brings about seismic performance assessment for the Cyprus transportation network from which the decision-making platform can be modeled and implemented. This study employs a seismic hazard analysis based on generated USGS ShakeMap scenarios for the risk assessment of the transportation network. Furthermore, identification of the resiliency and vulnerability of the transportation road network is carried out by utilizing the graph theory concept at the network level. Moreover, link performance measures, i.e., traffic modeling of the study region is simulated in a dynamic traffic assignment (DTA) simulation environment. Finally, for earthquake loss analysis of the bridges, the HAZUS loss estimation tool is used. The results of our investigations for three different earthquake scenarios have shown that seismic retrofitting of bridges is a cost-effective measure to reduce the structural and operational losses in the region.

On the dominant local metric dimension of some planar graphs

Dominant local metric dimension (DLMD) combines two interesting graph theory concepts: domination and local metric dimension (LMD). Graph invariants are a fantastic tool for analyzing graph abstract structures. Metric dimension and DLMD as graph invariants have a wide range of applications, which includes robot navigation, pharmaceutical chemistry, pattern recognition, and image processing. In this paper, we computed the DLMD of the prism graph [Formula: see text], antiprism graph [Formula: see text], and the sunflower graph [Formula: see text].

An algorithm for deriving the topology of belowground urban stormwater networks

Abstract. Belowground urban stormwater networks (BUSNs) are critical for removing excess rainfall from impervious urban areas and preventing or mitigating urban flooding. However, available BUSN data are sparse, preventing the modeling and analysis of urban hydrologic processes at regional and larger scales. We propose a novel algorithm for estimating BUSNs by drawing on concepts from graph theory and existing, extensively available land surface data, such as street network, topography, and land use/land cover. First, we derive the causal relationships between the topology of BUSNs and urban surface features based on graph theory concepts. We then apply the causal relationships and estimate BUSNs using web-service data retrieval, spatial analysis, and high-performance computing techniques. Finally, we validate the derived BUSNs in the metropolitan areas of Los Angeles, Seattle, Houston, and Baltimore in the US, where real BUSN data are partly available to the public. Results show that our algorithm can effectively capture 59 %–76 % of the topology of real BUSN data, depending on the supporting data quality. This algorithm has promising potential to support large-scale urban hydrologic modeling and future urban drainage system planning.

On k-regular edge connectivity of chemical graphs

Abstract Quantitative structure property research works, which are the essential part in chemical information and modelling, give basic underlying topological properties for chemical substances. This information enables conducting more feasible studies between theory and practice. Connectivity concept in chemical graph theory gives information about underlying topology of chemical structures, fault tolerance of molecules, and vulnerability of chemical networks. In this study we first defined two novel types of conditional connectivity measures based on regularity notion: k-regular edge connectivity and almost k-regular edge connectivity in chemical graph theory literature. We computed these new graph invariants for cycles, complete graphs, and Cartesian product of cycles. Our results will be applied to calculate k-regular edge connectivity of some nanotubes which are stated as Cartesian product of cycles. These calculations give information about fault tolerance capacity and vulnerability of these chemical structures.

Distributed Averaging Problems of Agriculture Picking Multi-Robot Systems via Sampled Control.

Distributed control of agriculture picking multi-robot systems has been widely used in the field of smart agriculture, this paper aims to explore the distributed averaging problems of agriculture picking multi-robot systems under directed communication topologies by taking advantage of the sampled data. With the algebraic graph theory concepts and the matrix theory, a distributed protocol is proposed based on the nearest sampled neighbor information. It is shown that under the proposed protocol, the states of all agents can be guaranteed to reach average consensus whose value is the averaging of the initial states of all agents. Besides, when considering time-delay, the other distributed protocol is constructed, in which a time margin of the time-delay can be determined simultaneously. The necessary and sufficient consensus results can be developed even though the time delay exists. Simulation results are given to demonstrate the effectiveness of our developed consensus results.

The Domination Parameters on a kind of the regular honeycomb structure

The honeycomb mesh, based on hexagonal structure, has enormous applications in chemistry and engineering. A major challenge in this field is to understand the unique properties of honeycomb structures, which depend on their properties of topology. One of the important concepts in graph theory is the domination number which can be used for network control and monitoring. In this paper, we investigate the domination number of the honeycomb network. For this purpose, the domination number, the total domination number, the independent domination number, the connected domination number and the doubly connected domination number of the honeycomb are obtained. Finally, in some honeycomb structures of real models, we obtain the exact amount of these parameters.

Concept of Temporal Pretopology for the Analysis for Structural Changes

Pretopology is a mathematical model developed from a weakening of the topological axiomatic. It was initially used in economic, social and biological sciences and next in pattern recognition and image analysis. More recently, it has been applied to the analysis of complex networks. Pretopology enables to work in a mathematical framework with weak properties, and its nonidempotent operator called pseudo-closure permits to implement iterative algorithms. It proposes a formalism that generalizes graph theory concepts and allows to model problems universally. In this paper, authors will extend this mathematical model to analyze complex data with spatiotemporal dimensions. Authors define the notion of a temporal pretopology based on a temporal function. They give an example of temporal function based on a binary relation, and construct a temporal pretopology. They define two new notions of temporal substructures which aim at representing evolution of substructures. They propose algorithms to extract these substructures. They experiment the proposition on 2 data and two economic real data.

Understanding the interaction among enablers of quality enhancement of higher business education in Pakistan.

This study aims to generate a list of enablers of quality enhancement of higher business education in Pakistan and build a structural model of enablers to prioritize them. It also intends to impose direction and hierarchy on the inter-relationships of the enablers. The study’s design consists of a literature review, data collection from primary sources, and qualitative analysis. Interpretive Structural Modeling (ISM) coupled with Matriced’ Impacts Cruise’s Multiplication Appliquée a UN Classement (MICMAC) is used as a research methodology. The classical procedure of ISM and MICMAC is applied to primary data collected by a field survey from a panel of experts recruited from folks of stakeholders of business education. Results of the literature show that eighteen critical enablers enhance the quality of higher business education in Pakistan. Results of ISM show that the enabler ’job placement of graduates’ occupies the top-level of the ISM model being least critical. In contrast, the enabler ’intra-academia linkages’ occupying the bottom of the model is the most vital. Results of MICMAC show that all enablers, except ’job placement of graduates, are classified into linkage clusters, whereas ‘job placement of graduates’ is classified as an independent cluster. Overall results of the study show that enablers of quality enhancement of higher business education in Pakistan are agile and not settled. The study has profound theoretical, managerial, and practical implications for all stakeholders of business education. It also provides a research framework for future studies concerning subject phenomena. The discussion about the structural model culminates into policy guidelines for the regulators. The study is subject to some methodological/data/resources limitations like the limited review of literature, collection of data from a medium-size panel of experts from Pakistan only, using majority rule for aggregating responses, answering only that what is related to what, other common limitations of qualitative studies, shot period and absence of financial support. The authors conduct this study in a real-life field setting is built on the original dataset and address the efficient issue of phenomenon understudy differently. It is theory-building research, therefore, does not require prior theory. It exploits simple elementary concepts of Boolean algebra, set theory, and graph theory that generates new in-depth information for stakeholders.

Optimal connected subgraphs: Integer programming formulations and polyhedra

AbstractConnectivity is a central concept in combinatorial optimization, graph theory, and operations research. In many applications, one is interested in finding an optimal subset of vertices with the essential requirement that the vertices are connected, but not how they are connected. In other words, it is not relevant which edges are selected to obtain connectivity. This article is concerned with the exact solution of such problems via integer programming. We analyze and compare (mixed) integer programming formulations with respect to the strength of their linear programming relaxations. Along the way, we also provide a tighter (compact) description of the connected subgraph polytope—the convex hull of subsets of vertices that induce a connected subgraph. Furthermore, we give a (compact) complete description of the connected subgraph polytope for graphs with no four independent vertices.

Variance and Covariance of Distributions on Graphs

We develop a theory to measure the variance and covariance of probability distributions defined on the nodes of a graph, which takes into account the distance between nodes. Our approach generalizes the usual (co)variance to the setting of weighted graphs and retains many of its intuitive and desired properties. Interestingly, we find that a number of famous concepts in graph theory and network science can be reinterpreted in this setting as variances and covariances of particular distributions. As a particular application, we define the maximum variance problem on graphs with respect to the effective resistance distance, and we characterize the solutions to this problem both numerically and theoretically. We show how the maximum variance distribution is concentrated on the boundary of the graph, and illustrate this in the case of random geometric graphs. Our theoretical results are supported by a number of experiments on a network of mathematical concepts, where we use the variance and covariance as analytical tools to study the (co)occurrence of concepts in scientific papers with respect to the (network) relations between these concepts.

Neural Correlation of Faradarmani Consciousness Field Mind Mediation: A Comparative Functional Connectivity and Graph analysis

The study of brain networks using analysis of electroencephalography (EEG) data based on statistical dependencies (functional connectivity) and mathematical graph theory concepts are common in neuroscience and cognitive sciences for examinations of patients and healthy individuals. Taheri Consciousness Fields and their applications in the optimization of the systems under study have been investigated in various studies. In this study, we examined the results of applying the Faradarmani Consciousness Field (CF) in the Faradarmangars’ brains (a certified and trained individual who has been entrusted with the TCFs). According to Taheri, the effects of Faradarmani CF are initiated through Faradarmangars’ minds. For this purpose, the functional and effective connectivity, and the corresponding brain graphs of EEG from the brains of a group of Faradarmangar are compared with that of non-Faradarmangar groups during Faradarmani CF Connection. According to the results, the brain of the Faradarmangars showed a significantly decreased activity in delta (BA8), beta2 (BA4/6/8/9/10/11/32/44/47), and beta3 (in 34 of 52 BA) frequency bands, mainly in the frontal lobe and after that in parietal and temporal lobes in comparison with the non-Faradarmangars. Moreover, the frontal network’s functional and effective connectivity analysis showed dominant multiple decreased connectivity, mainly in the case of the beta3 frequency band in all parts of the frontal network. On the other hand, the graph theory analysis of the Faradarmangar brain indicated an increase in the activity of the O2-T5-F4-F3-FP2-F8 areas and a significant decrease in the characteristic path length and increases in global efficiency, clustering coefficient and transitivity. In conclusion, the unique higher graph function efficiency and the reduction in the brain activity and connectivity during the Faradarmani CF mind mediation showed the human brain's passive and detector-like function in this task.

The local complement metric dimension of graphs

The metric dimension of graphs has been extended in some types and variations such as local metric dimension and complement metric dimension of graphs. Merging two concepts is one way of developing the concept of graph theory as a branch of mathematics. These two variations motivated us to construct a new concept of metric dimension so called local complement metric dimension. We apply this new concept to some particular classes of graphs and the corona product of two graphs. We also characterize the local complement metric dimension of graphs with certain properties, namely, bipartite graphs and odd cycle graphs. Furthermore, we discover the local complement metric dimension of the corona product of two particular graphs as well as the corona product of two general graphs.

Dynamical and structural study of a generalized Caputo fractional order Lotka-Volterra model

In this study, we generalize the Lotka-Volterra (LV) model. Our generalization is done via two simultaneous techniques. The first technique is done by incorporating a general term to model the immigrations to the predator or prey populace. The second technique is to utilize the Caputo fractional derivative. This study is done via different five cases. We show the impact of changing the fractional order to the proposed model response. We display that including a generalized immigration to the predator or prey populace produces stabilization of the LV model in context of Caputo fractional derivatives in all executed cases. In addition, we present the generalized model via graph and utilizing the graph theory concepts, we display several hidden features of the model.

A study of applications of graph colouring in various fields

Graph theory is an important branch of applied mathematics with a lot of applications in many fields. Graph theory has a broad scale of applications in many practical situations. Graph coloring is one decent approach which can deal with many problems of graph theory. The main aim of this paper is to present the importance of graph coloring ideas in various fields for researchers that the concept of graph theory can be used by them. In this paper an overview is presented especially to project the idea of graph theory and applications of graph colouring.

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.