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Journal articles on the topic 'Graph theory. Categories (Mathematics)'

Author: Grafiati

Published: 4 June 2021

Last updated: 13 February 2022

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1

Medová, Páleníková, Rybanský, and Naštická. "Undergraduate Students’ Solutions of Modeling Problems in Algorithmic Graph Theory." Mathematics 7, no. 7 (June 26, 2019): 572. http://dx.doi.org/10.3390/math7070572.

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Graphs can be considered as useful mathematical models. Graph algorithms are a common part of undergraduate courses in discrete mathematics. Even though they have been successfully implemented in secondary curricula, little research has been dedicated to the analysis of students’ work. Within a discrete mathematics course for university students, several graph algorithms were introduced via their applications. At the end of the course, the students took a test focused, inter alia, on applications of the algorithms. The mistakes that occurred in 127 students’ solutions of three problems (the Chinese postman problem, the shortest path problem, and the minimum spanning tree problem) were categorized and compared. Surprisingly, no mistakes were identified in the mathematization of situations or in the interpretation of results with respect to the wording of the problem. The categories of errors varied regardless of the problem types. Hierarchical cluster analysis grouped together the students’ solutions for the Chinese postman problem and the minimum spanning tree problem. By means of nonparametric item response theory analysis, the Chinese postman problem was identified as the most problematic for students. Possible sources of this difficulty are discussed in more detail herein.

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Grossman, Stuart C. "On graph theoretical SAR and the mathematical theory of categories." Journal of Molecular Structure: THEOCHEM 233 (September 1991): 21–34. http://dx.doi.org/10.1016/0166-1280(91)85051-8.

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Derksen, Harm. "The Graph Isomorphism Problem and approximate categories." Journal of Symbolic Computation 59 (December 2013): 81–112. http://dx.doi.org/10.1016/j.jsc.2013.06.002.

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Phoa, Wesley. "Building domains from graph models." Mathematical Structures in Computer Science 2, no. 3 (September 1992): 277–99. http://dx.doi.org/10.1017/s0960129500001481.

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In this paper we study partial equivalence relations (PERs) over graph models of the λcalculus. We define categories of PERs that behave like predomains, and like domains. These categories are small and complete; so we can solve domain equations and construct polymorphic types inside them. Upper, lower and convex powerdomain constructions are also available, as well as interpretations of subtyping and bounded quantification. Rather than performing explicit calculations with PERs, we work inside the appropriate realizability topos: this is a model of constructive set theory in which PERs, can be regarded simply as special kinds of sets. In this framework, most of the definitions and proofs become quite smple and attractives. They illustrative some general technicques in ‘synthetic domain theory’ that rely heavily on category theory; using these methods, we can obtain quite powerful results about classes of PERs, even when we know very little about their internal structure.

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BRUGGINK, H. J. SANDER, and BARBARA KÖNIG. "Recognizable languages of arrows and cospans." Mathematical Structures in Computer Science 28, no. 8 (August 8, 2018): 1290–332. http://dx.doi.org/10.1017/s096012951800018x.

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In this article, we generalize Courcelle's recognizable graph languages and results on monadic second-order logic to more general structures.First, we give a category-theoretical characterization of recognizability. A recognizable subset of arrows in a category is defined via a functor into the category of relations on finite sets. This can be seen as a straightforward generalization of finite automata. We show that our notion corresponds to recognizable graph languages if we apply the theory to the category of cospans of graphs.In the second part of the paper, we introduce a simple logic that allows to quantify over the subobjects of a categorical object. Again, we show that, for the category of graphs, this logic is equally expressive as monadic second-order graph logic (msogl). Furthermore, we show that in the more general setting of hereditary pushout categories, a class of categories closely related to adhesive categories, we can recover Courcelle's result that everymsogl-expressible property is recognizable. This is done by giving an inductive translation of formulas of our logic into automaton functors.

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Ul Haq Bokhary, Syed Ahtsham, Muhammad Imran, Shehnaz Akhter, and Sadia Manzoor. "Molecular topological invariants of certain chemical networks." Main Group Metal Chemistry 44, no. 1 (January 1, 2021): 141–49. http://dx.doi.org/10.1515/mgmc-2021-0010.

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Abstract Topological descriptors are the graph invariants that are used to explore the molecular topology of the molecular/chemical graphs. In QSAR/QSPR research, physico-chemical characteristics and topological invariants including Randić, atom-bond connectivity, and geometric arithmetic invariants are utilized to corelate and estimate the structure relationship and bioactivity of certain chemical compounds. Graph theory and discrete mathematics have discovered an impressive utilization in the area of research. In this article, we investigate the valency-depended invariants for certain chemical networks like generalized Aztec diamonds and tetrahedral diamond lattice. Moreover, the exact values of invariants for these categories of chemical networks are derived.

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Ferri, Massimo. "Colour Switching and Homeomorphism of Manifolds." Canadian Journal of Mathematics 39, no. 1 (February 1, 1987): 8–32. http://dx.doi.org/10.4153/cjm-1987-002-5.

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Throughout this paper, we work in the PL and pseudosimplicial categories, for which we refer to [17] and [10] respectively. For the graph theory involved see [9].An h-coloured graph (Γ, γ) is a multigraph Γ = (V(Γ), E(Γ)) regular of degree h, endowed with an edge-coloration γ by h colours. If is the colour set, for each we setFor each set . For n ∊ Z, n ≧ 1, setΔn will be mostly used to denote the colour set for an (n + 1)-coloured graph.

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FOUNTOULAKIS, NIKOLAOS, and KONSTANTINOS PANAGIOTOU. "3-Connected Cores In Random Planar Graphs." Combinatorics, Probability and Computing 20, no. 3 (January 24, 2011): 381–412. http://dx.doi.org/10.1017/s0963548310000532.

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The study of the structural properties of large random planar graphs has become in recent years a field of intense research in computer science and discrete mathematics. Nowadays, a random planar graph is an important and challenging model for evaluating methods that are developed to study properties of random graphs from classes with structural side constraints.In this paper we focus on the structure of random 2-connected planar graphs regarding the sizes of their 3-connected building blocks, which we callcores. In fact, we prove a general theorem regarding random biconnected graphs from various classes. IfBnis a graph drawn uniformly at random from a suitable classof labelled biconnected graphs, then we show that with probability 1 −o(1) asn→ ∞,Bnbelongs to exactly one of the following categories:(i)either there is a uniquegiantcore inBn, that is, there is a 0 <c=c() < 1 such that the largest core contains ~cnvertices, and every other core contains at mostnαvertices, where 0 < α = α() < 1;(ii)or all cores ofBncontainO(logn) vertices.Moreover, we find the critical condition that determines the category to whichBnbelongs, and also provide sharp concentration results for the counts of cores of all sizes between 1 andn. As a corollary, we obtain that a random biconnected planar graph belongs to category (i), where in particularc= 0.765. . . and α = 2/3.

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Mordecai, Yaniv, James P. Fairbanks, and Edward F. Crawley. "Category-Theoretic Formulation of the Model-Based Systems Architecting Cognitive-Computational Cycle." Applied Sciences 11, no. 4 (February 23, 2021): 1945. http://dx.doi.org/10.3390/app11041945.

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We introduce the Concept→Model→Graph→View Cycle (CMGVC). The CMGVC facilitates coherent architecture analysis, reasoning, insight, and decision making based on conceptual models that are transformed into a generic, robust graph data structure (GDS). The GDS is then transformed into multiple views of the model, which inform stakeholders in various ways. This GDS-based approach decouples the view from the model and constitutes a powerful enhancement of model-based systems engineering (MBSE). The CMGVC applies the rigorous foundations of Category Theory, a mathematical framework of representations and transformations. We show that modeling languages are categories, drawing an analogy to programming languages. The CMGVC architecture is superior to direct transformations and language-coupled common representations. We demonstrate the CMGVC to transform a conceptual system architecture model built with the Object Process Modeling Language (OPM) into dual graphs and a stakeholder-informing matrix that stimulates system architecture insight.

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GARVER, ALEXANDER, and THOMAS MCCONVILLE. "ORIENTED FLIP GRAPHS, NONCROSSING TREE PARTITIONS, AND REPRESENTATION THEORY OF TILING ALGEBRAS." Glasgow Mathematical Journal 62, no. 1 (February 7, 2019): 147–82. http://dx.doi.org/10.1017/s0017089519000028.

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AbstractThe purpose of this paper is to understand lattices of certain subcategories in module categories of representation-finite gentle algebras called tiling algebras, as introduced by Coelho Simões and Parsons. We present combinatorial models for torsion pairs and wide subcategories in the module category of tiling algebras. Our models use the oriented flip graphs and noncrossing tree partitions, previously introduced by the authors, and a description of the extension spaces between indecomposable modules over tiling algebras. In addition, we classify two-term simple-minded collections in bounded derived categories of tiling algebras. As a consequence, we obtain a characterization of c-matrices for any quiver mutation-equivalent to a type A Dynkin quiver.

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Maley, Carlo C. "DNA Computation: Theory, Practice, and Prospects." Evolutionary Computation 6, no. 3 (September 1998): 201–29. http://dx.doi.org/10.1162/evco.1998.6.3.201.

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L. M. Adleman launched the field of DNA computing with a demonstration in 1994 that strands of DNA could be used to solve the Hamiltonian path problem for a simple graph. He also identified three broad categories of open questions for the field. First, is DNA capable of universal computation? Second, what kinds of algorithms can DNA implement? Third, can the error rates in the manipulations of the DNA be controlled enough to allow for useful computation? In the two years that have followed, theoretical work has shown that DNA is in fact capable of universal computation. Furthermore, algorithms for solving interesting questions, like breaking the Data Encryption Standard, have been described using currently available technology and methods. Finally, a few algorithms have been proposed to handle some of the apparently crippling error rates in a few of the common processes used to manipulate DNA. It is thus unlikely that DNA computation is doomed to be only a passing curiosity. However, much work remains to be done on the containment and correction of errors. It is far from clear if the problems in the error rates can be solved sufficiently to ever allow for general-purpose computation that will challenge the more popular substrates for computation. Unfortunately, biological demonstrations of the theoretical results have been sadly lacking. To date, only the simplest of computations have been carried out in DNA. To make significant progress, the field will require both the assessment of the practicality of the different manipulations of DNA and the implementation of algorithms for realistic problems. Theoreticians, in collaboration with experimentalists, can contribute to this research program by settling on a small set of practical and efficient models for DNA computation.

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Wiweger, Antoni. "Free Games over Coloured Automata." Fundamenta Informaticae 8, no. 2 (April 1, 1985): 199–224. http://dx.doi.org/10.3233/fi-1985-8204.

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Concepts of category theory are applied to the investigation of some relations between automata and abstract games. The notion of a coloured automaton introduced in this paper provides a framework for a unified treatment of automata and abstract games. Both games and automata can be viewed as special cases of this general notion. A coloured automaton is defined to be a Mealy automaton with the additional structure of a coloured graph on the set of inputs. Various categories of coloured automata, automata, and games are described. It is shown that some forgetful functors between these categories have left adjoints, and explicit constructions of these adjoints are given. The main result is Theorem 5.5 which describes a construction of a free abstract game over a coloured automaton satisfying some additional conditions.

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Mamunts, D., S. Sokolov, A. Nyrkov, S. Chernyi, M. Bukhurmetov, and V. Kuznetsov. "Models and Algorithms for Estimation and Minimization of the Risks Associated with Dredging." Transport and Telecommunication Journal 18, no. 2 (June 1, 2017): 139–45. http://dx.doi.org/10.1515/ttj-2017-0013.

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Abstract There are a lot of models and algorithms to minimize risks during dredging operations and they are not without drawbacks. The paper describes the authors’ approach to solving this problem. Mathematical models are proposed and on their basis software is developed. Methods of the risk theory are used to minimize the risks. In this paper a consequence of influence refers to the deviation from the goal expressed in the expected results and the deviation of certain criterion factors. In this case, we mean any measure of quality. In its turn, risk factors reduce criterion factors. These factors are divided into categories - general transportation risks and risks of transporting ground. In these categories, one may derive the following risks - incidents at transport resulting from the impact of a set of random factors including the human one. For risk analysis and management, in addition to identifying critical chains of risk situations, the stochastic model for evaluating the chains is set forth. In order to implement this algorithm, the mathematical package Maple is used, which allows for conducting the required calculations with a software package including the Graph Theory. The paper presents fragments of the code listing.

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Saitov, Igor', Anna Motienko, Sergei Astapov, and Oleg Basov. "Synthesis of the Topological Structure of Distributed Terminal System for Audio Monitoring of Users of Local Information Spaces." SPIIRAS Proceedings 18, no. 6 (November 29, 2019): 1357–80. http://dx.doi.org/10.15622/sp.2019.18.6.1357-1380.

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A widespread use of multi-user interfaces, due to multimodality of traditional interpersonal communication, a transition to a polymerized presentation of information and systems, has allowed the creation of new approaches to their implementation based on distributed terminal systems. An approach to the synthesis of topological structures of such systems implemented in two stages is proposed in the article. The first stage determines a minimum set of communication nodes and their location based on the requirements for the availability of communication nodes for various categories of users and the globality of a distributed terminal system. The second stage determines options for constructing communication nodes and connections between them, which ensure the performance of audio monitoring functions of users of local information spaces while ensuring continuity of a bridge for different categories of users. A model example of the synthesis of a distributed terminal system for audio monitoring of two categories of users (adults and children) in the local information space (home), voice control subsystems of the "smart home" is presented. As a part of its solution, at each stage of the synthesis, the initial data are determined, a formal formulation of the synthesis problem is carried out, an algorithm for the solution and the results are presented. So the task of the first stage of the synthesis is a linear integer mathematical programming problem, solved in the model example by the simplex method, the solution of the second stage problem is based on the alternative graph formalization and the method of "branches and borders". The obtained results clearly demonstrate the capabilities of the proposed scientific and methodological tools for the synthesis of the topological structure of distributed terminal systems and the prospects of its use in the newly arising tasks of the technical implementation of new infocommunication technologies and services.

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DA FONTOURA COSTA, LUCIANO. "WHAT'S IN A NAME?" International Journal of Modern Physics C 15, no. 03 (March 2004): 371–79. http://dx.doi.org/10.1142/s0129183104005772.

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Among the several findings deriving from the application of complex network formalism to the investigation of natural phenomena, the fact that linguistic constructions follow power laws presents special interest for its potential implications for psychology and brain science. By corresponding to one of the most essentially human manifestations, such language-related properties suggest that similar dynamics may also be inherent to the brain areas related to language and associative memory, and perhaps even consciousness. The present work reports a preliminary experimental investigation aimed at characterizing and modeling the flow of sequentially induced associations between words from the English language in terms of complex networks. The data is produced through a psychophysical experiment where a word is presented to the subject, who is requested to associate another word. Complex network and graph theory formalism and measurements are applied in order to characterize the experimental data. Several interesting results are identified, including the characterization of attraction basins, association asymmetries, context biasing, as well as a possible power-law underlying word associations, which could be explained by the appearance of strange loops along the hierarchical structure underlying word categories.

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Kopytko, Marta, Anna Pazieieva, Andrii Khorosheniuk, Mykhailo Matviienko, and Мariya Vinichuk. "SHADOW EMPLOYMENT IN EASTERN EUROPE: PRACTICAL ASPECTS OF EVALUATION AND COUNTERACTION." Business: Theory and Practice 20 (December 17, 2019): 485–91. http://dx.doi.org/10.3846/btp.2019.45.

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The paper outlines the main problems of shadowing of the labor market in Eastern Europe on the example of Ukraine and explores the factors that have a direct impact on them. A comparative analysis of the most widespread methods of tax evasion in Ukraine has been conducted. Using the software package MS Excel and CurveExpert 5.0, a study was made of the dynamics of the population employed in the informal economy during the period from 2001 to 2018; the impact of the level of labor market shadowing on the standard of living of the population of Ukraine, as well as the dynamics of the volume of remittances of labor migrants to Ukraine. The level of employment of the population in the informal sector of the economy is analyzed. It is proposed one of the alternative methods of economical and mathematical nature, with the help of which it is possible to conduct researches of the imbalance of economic systems, the construction of a mathematical model of balance on the basis of modeling of a small group using the graph theory, which will enable to carry out tactical and strategic control over the interaction of the main categories of the population in the labor market.

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Karim, Abdul, Azhari Azhari, Samir Brahim Belhaouri, Ali Adil Qureshi, and Maqsood Ahmad. "Methodology for Analyzing the Traditional Algorithms Performance of User Reviews Using Machine Learning Techniques." Algorithms 13, no. 8 (August 18, 2020): 202. http://dx.doi.org/10.3390/a13080202.

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Android-based applications are widely used by almost everyone around the globe. Due to the availability of the Internet almost everywhere at no charge, almost half of the globe is engaged with social networking, social media surfing, messaging, browsing and plugins. In the Google Play Store, which is one of the most popular Internet application stores, users are encouraged to download thousands of applications and various types of software. In this research study, we have scraped thousands of user reviews and the ratings of different applications. We scraped 148 application reviews from 14 different categories. A total of 506,259 reviews were accumulated and assessed. Based on the semantics of reviews of the applications, the results of the reviews were classified negative, positive or neutral. In this research, different machine-learning algorithms such as logistic regression, random forest and naïve Bayes were tuned and tested. We also evaluated the outcome of term frequency (TF) and inverse document frequency (IDF), measured different parameters such as accuracy, precision, recall and F1 score (F1) and present the results in the form of a bar graph. In conclusion, we compared the outcome of each algorithm and found that logistic regression is one of the best algorithms for the review-analysis of the Google Play Store from an accuracy perspective. Furthermore, we were able to prove and demonstrate that logistic regression is better in terms of speed, rate of accuracy, recall and F1 perspective. This conclusion was achieved after preprocessing a number of data values from these data sets.

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Lapinova, Svetlana, Alena Anikina, and Alexander Osharin. "Analysis of export and import structures using network methods (on the example of the agricultural market)." St Petersburg University Journal of Economic Studies 36, no. 3 (2020): 421–54. http://dx.doi.org/10.21638/spbu05.2020.304.

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Analysis of trade cooperation between countries and identification of the most significant market participants is of great importance, both theoretically and empirically. The global trading community forms a network of international relations defined by trade contracts in various industries. Export-import trade flows are one of the key indicators of the level of cooperation among countries and the state of the global economy. The high intensity of such contacts across groups of countries suggests the existence of clusters in this market segment,consisting of central players — exporters and importers, who often define rules for other participants.Understanding the existence and identification of such a center helps to develop an optimal international trade strategy. The purpose of this contribution is to identify factors affecting trade flows among different countries. Statistical analysis of the international trade relations does not always reveal all the essential aspects of cooperation. This paper combines the methods of graph theory and econometric analysis to study the parameters of trade flows among countries. The parameters used in the network analysis make it possible to obtain additional characteristics of market participants, which help to evaluate their significance in the world trade. The paper also identifies some key mathematical and economic characteristics of export-import flows connecting destination countries. We have analyzed the directions of changes in world trade and established correspondences between metric characteristics of graph vertices and parameters of world trade models. The Russian indicators in export/import categories and its largest sales agents are estimated. The identification of the key intermediaries and importers (centers and authorities) on each of the markets in question has been carried out. As an example for this identification the market of agricultural products among the world’s largest exporters and importers of the product were used.

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Frölich, Thomas, F. F. Bevier, Alicja Babakhani, Hannah H. Chisholm, Peter Henningsen, David S. Miall, Seija Sandberg, and Argogast Schmitt. "The ontology of person-centered healthcare: Theoretical essentials to reground medicine within its humanistic framework. Part I - A centered concept of personhood: Some tools to conceptualise subjectivity." European Journal for Person Centered Healthcare 6, no. 1 (April 30, 2018): 146. http://dx.doi.org/10.5750/ejpch.v6i1.1375.

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To address subjectivity, as a generally rooted phenomenon, other ways of visualisation must be applied than in conventional objectivistic approaches. Using ‘trees’ as operational metaphors, as employed in Arthur Cayley’s ‘theory of the analytical forms called trees’, one rooted ‘tree’ must be set beneath the other and, if such ‘trees’ are combined, the resulting ‘forest’ is nevertheless made up of individual ‘trees’ and not of a deconstructed mix of ‘roots’, ‘branches’, ‘leaves’ or further categories, each understood as addressable both jointly and individually. The reasons for why we have chosen a graph theory and corresponding discrete mathematics as an approach and application are set out in this first of our three articles. It combines two approaches that, in combination, are quite uncommon and which are therefore not immediately familiar to all readers. But as simple as it is to imagine a tree, or a forest, it is equally simple to imagine a child blowing soap bubbles with the aid of a blow ring. A little more challenging, perhaps, is the additional idea of arranging such blow rings in series, transforming the size of the soap bubble in one ring after the other. To finally combine both pictures, the one of trees and the other of blow rings, goes beyond simple imagination, especially when we prolong the imagined blow ring becoming a tunnel, with a specific inner shape. The inner shape of the blow ring and its expansion as a tunnel are understood as determined by discrete qualities, each forming an internal continuity, depicted as a scale, with the scales combined in the form of a glyph plot. The different positions on these scales determine their length and if the endpoints of the spines are connected with an enveloping line then this corresponds to the free space left open in the tunnel to go through it. Using so many visualisation techniques at once is testing. Nevertheless, this is what we propose here and to facilitate such a visualisation within the imagination, we do it step by step. As the intended result of this ‘juggling of three balls’, as it were, we end up with a concept of how living beings elaborate their principal structure to enable controlled outside-inside communication.

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Muhammad Ahlami Ashraf Roslan, Murizah Kassim,. "Analysisof Students’ Web Browsing Behaviours Using Data Miningat a Campus Network." Turkish Journal of Computer and Mathematics Education (TURCOMAT) 12, no. 6 (April 5, 2021): 2726–38. http://dx.doi.org/10.17762/turcomat.v12i6.5779.

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Analytics provides insight to people based on the analytics of past usage by using techniques such as statistics, data mining, machine learning and artificial intelligence. Lack of monitoring system of browsing causes low engagements that reduce the growth of certain businesses caused by unnecessary browsing for students learning time. This paper presents an analysis on browsing behavior that classifies browsed words followed their ethical word-groups browsing. An Analytic platform is created as a monitoring system of browsing behavior. Data mining, indexing and classification method are used in this research as data is the essential key of creating a predictive model and four types of ethical groups have been filtered based on the browsing behaviors. The browsed words are categorized into four types of browsing called queries, applications, social media, Campus-related sites. The research method uses software tools and data mining process on the browsing data and analytics is presented on the development of the dashboard mainly using the R programming language. Few unethical words using the indexing method are generated in analytic graphs based on the type of browsing versus time. Data collected from the browsing behaviors of students’analysis taken from browsing database of personal computer and laboratory computer in the campus network. The result shows that othercategories are the highest categories which reached79.6% for personals' computer browsing compared to72.4% browsing at the laboratory computers. It is identified that about 21% of the browsing behavior was filtered during the data mined processed. The other category is still on the research portfolio where these libraries must be filtered in detail to identify whether they are learning or non-learning activities. This research is significant in that helps to increase the effectiveness of suggestions applications, optimize the internet usage by blocking unnecessary words or webpages, and even campus guide systems by monitoring the surrounding browsing behavior of the students’ usages of the campus network computer labs.

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Shalamberidze, Irakli, and Merab Akhobadze. "Web platform for "Smart City" data collection and analytics." ECONOMIA AGRO-ALIMENTARE, no. 3 (January 2020): 847–54. http://dx.doi.org/10.3280/ecag2019-003015.

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The study aims to highlight that nowadays, finding ways to manage the current processes both in the regions and in cities with big agglomeration is the most important and difficult problem. A fortiori, when it concerns developed regions. While designing urban system development, management, and reconstruction projects, both managers of the cities and urbanists must take into account the opinions of specialists, who have different categories of mindsets and they "talk different languages" (Sociologists, ecologists, businessmen, etc.). Summing up the aforementioned languages in a common denominator is possible only by mathematics and computing tools. Nowadays, the problems of city management are united in the concept of "Smart city", which is usually referred to as "informational city". "Smart City" - this is an integration concept, which involves the usage of the so called "integrated imitative model" for systematic, stable, optimal decision making, as the city is a whole dynamic unity. Today's managers of the cities, urbanists, investors, businessmen, sociologists, etc. have to deal with a huge amount of parameters, opinions and data in a nonsystematic manner. Our proposed study "Unified Web Platform of the Region and Smart Management" includes: website, Google Map, pointing object in the map, saving the objects and their parameters, mathematical and programmatic tools, cloud computing, python computing libraries, Restful api as a web service, etc. As for the web service or restful api, any software can have access to the data of the united web platform of the region through a specially defined protocol. Objects presented in the map have assigned specialized and standardized parameters, which are used by the system algorithm for the analyses and the presentation of all the structural creators of the dynamic processes of the city. This gives us the opportunity to see the whole chain of interactions, which are caused by the actions on any object of the city. Users register on the website and they can see the parameters of the objects that are set in the map. The objects in the databases are classified by their purpose, affiliation, destination and other marks. There is an ability for users to define the status of an object on their own. Users can also add or remove objects on the map and can manipulate with the updated parameters on the map. They can evaluate the chain of results both in the time and dimensional manner. For the built-in mathematical tools and algorithms in the system, we use Algebraic topology methods, Graphs theory non-linear differential equations, the theory of disasters and bifurcation, Chaos theory, methods of mathematical statistics and more. Web platform includes all the mathematical tools and programmatic packages that are necessary for stable development of small and medium-sized business.

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Alborzi, Seyed Ziaeddin, Amina Ahmed Nacer, Hiba Najjar, David W. Ritchie, and Marie-Dominique Devignes. "PPIDomainMiner: Inferring domain-domain interactions from multiple sources of protein-protein interactions." PLOS Computational Biology 17, no. 8 (August 9, 2021): e1008844. http://dx.doi.org/10.1371/journal.pcbi.1008844.

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Many biological processes are mediated by protein-protein interactions (PPIs). Because protein domains are the building blocks of proteins, PPIs likely rely on domain-domain interactions (DDIs). Several attempts exist to infer DDIs from PPI networks but the produced datasets are heterogeneous and sometimes not accessible, while the PPI interactome data keeps growing. We describe a new computational approach called “PPIDM” (Protein-Protein Interactions Domain Miner) for inferring DDIs using multiple sources of PPIs. The approach is an extension of our previously described “CODAC” (Computational Discovery of Direct Associations using Common neighbors) method for inferring new edges in a tripartite graph. The PPIDM method has been applied to seven widely used PPI resources, using as “Gold-Standard” a set of DDIs extracted from 3D structural databases. Overall, PPIDM has produced a dataset of 84,552 non-redundant DDIs. Statistical significance (p-value) is calculated for each source of PPI and used to classify the PPIDM DDIs in Gold (9,175 DDIs), Silver (24,934 DDIs) and Bronze (50,443 DDIs) categories. Dataset comparison reveals that PPIDM has inferred from the 2017 releases of PPI sources about 46% of the DDIs present in the 2020 release of the 3did database, not counting the DDIs present in the Gold-Standard. The PPIDM dataset contains 10,229 DDIs that are consistent with more than 13,300 PPIs extracted from the IMEx database, and nearly 23,300 DDIs (27.5%) that are consistent with more than 214,000 human PPIs extracted from the STRING database. Examples of newly inferred DDIs covering more than 10 PPIs in the IMEx database are provided. Further exploitation of the PPIDM DDI reservoir includes the inventory of possible partners of a protein of interest and characterization of protein interactions at the domain level in combination with other methods. The result is publicly available at http://ppidm.loria.fr/.

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Catlin, Paul A., and Gary Chartrand. "Introductory Graph Theory." American Mathematical Monthly 94, no. 5 (May 1987): 483. http://dx.doi.org/10.2307/2322751.

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Wilson, B. J. "ALGORITHMIC GRAPH THEORY." Bulletin of the London Mathematical Society 18, no. 6 (November 1986): 630–31. http://dx.doi.org/10.1112/blms/18.6.630.

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Jiang, Yaozhi. "Fractal graph and integral graph: C-type hyper graph in graph theory." Applied Mathematical Sciences 13, no. 3 (2019): 135–49. http://dx.doi.org/10.12988/ams.2019.812193.

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Harary, Frank. "Graph theory as applied mathematics." Journal of Graph Theory 10, no. 3 (1986): iii—iv. http://dx.doi.org/10.1002/jgt.3190100303.

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Brightwell, G. "GRAPH DECOMPOSITIONS-A STUDY IN INFINITE GRAPH THEORY." Bulletin of the London Mathematical Society 24, no. 1 (January 1992): 90–92. http://dx.doi.org/10.1112/blms/24.1.90.

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Olver, Peter J., and Chehrzad Shakiban. "Graph theory and classical invariant theory." Advances in Mathematics 75, no. 2 (June 1989): 212–45. http://dx.doi.org/10.1016/0001-8708(89)90038-8.

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Gasarch, William, and Jeffry L. Hirst. "Reverse Mathematics and Recursive Graph Theory." Mathematical Logic Quarterly 44, no. 4 (1998): 465–73. http://dx.doi.org/10.1002/malq.19980440405.

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ANDERSON, OLIVER D. "Graph Theory Mind-Grinders." Teaching Mathematics and its Applications 10, no. 2 (1991): 74–81. http://dx.doi.org/10.1093/teamat/10.2.74.

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Albertson, Michael O. "Book Review: Modern graph theory." Bulletin of the American Mathematical Society 36, no. 03 (April 27, 1999): 389–91. http://dx.doi.org/10.1090/s0273-0979-99-00781-8.

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Thomassen, Carsten. "Book Review: Topological graph theory." Bulletin of the American Mathematical Society 19, no. 2 (October 1, 1988): 560–62. http://dx.doi.org/10.1090/s0273-0979-1988-15742-4.

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Amit, Alon, and Nathan Linial. "Random Graph Coverings I: General Theory and Graph Connectivity." Combinatorica 22, no. 1 (January 1, 2002): 1–18. http://dx.doi.org/10.1007/s004930200000.

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Nadler, Sam B. "Continuum Theory and Graph Theory: Disconnection Numbers." Journal of the London Mathematical Society s2-47, no. 1 (February 1993): 167–81. http://dx.doi.org/10.1112/jlms/s2-47.1.167.

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Hoshino, Richard, and Ken-ichi Kawarabayashi. "Graph Theory and Sports Scheduling." Notices of the American Mathematical Society 60, no. 06 (January 1, 2013): 1. http://dx.doi.org/10.1090/noti1010.

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Sebő, András, and Zoltán Szigeti. "Preface: Graph theory and combinatorics." Discrete Applied Mathematics 209 (August 2016): 1. http://dx.doi.org/10.1016/j.dam.2016.02.021.

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Pendavingh, Rudi, Quintijn Puite, and Gerhard J. Woeginger. "2-piercings via graph theory." Discrete Applied Mathematics 156, no. 18 (November 2008): 3510–12. http://dx.doi.org/10.1016/j.dam.2008.03.026.

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Manacher, Glenn K. "Algorithmic Graph Theory (Alan Gibbons)." SIAM Review 31, no. 1 (March 1989): 145–47. http://dx.doi.org/10.1137/1031028.

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Bukh, Boris, and David Conlon. "Rational exponents in extremal graph theory." Journal of the European Mathematical Society 20, no. 7 (May 22, 2018): 1747–57. http://dx.doi.org/10.4171/jems/798.

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Hopkins, Brian. "KEVIN BACON AND GRAPH THEORY." PRIMUS 14, no. 1 (January 2004): 5–11. http://dx.doi.org/10.1080/10511970408984072.

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Stein, Maya. "Extremal infinite graph theory." Discrete Mathematics 311, no. 15 (August 2011): 1472–96. http://dx.doi.org/10.1016/j.disc.2010.12.018.

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LIU, XIAOGANG. "SELECTED TOPICS IN SPECTRAL GRAPH THEORY." Bulletin of the Australian Mathematical Society 93, no. 3 (February 17, 2016): 511–12. http://dx.doi.org/10.1017/s0004972715001768.

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Arafa, N. A., M. Shokry, and M. Hassan. "From graph theory to nano topology." Filomat 34, no. 1 (2020): 1–17. http://dx.doi.org/10.2298/fil2001001a.

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Musy, François, Laurent Nicolas, and Ronan Perrussel. "Gradient-prolongation commutativity and graph theory." Comptes Rendus Mathematique 341, no. 11 (December 2005): 707–12. http://dx.doi.org/10.1016/j.crma.2005.09.037.

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Backman, Spencer. "Riemann–Roch theory for graph orientations." Advances in Mathematics 309 (March 2017): 655–91. http://dx.doi.org/10.1016/j.aim.2017.01.005.

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White, Arthur T. "Topological Graph Theory: A Personal Account." Electronic Notes in Discrete Mathematics 31 (August 2008): 5–15. http://dx.doi.org/10.1016/j.endm.2008.06.014.

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Koster, Arie, and Vadim Lozin. "DIMAP Workshop on Algorithmic Graph Theory." Electronic Notes in Discrete Mathematics 32 (March 2009): 1. http://dx.doi.org/10.1016/j.endm.2009.02.001.

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Christensen, J. Daniel, William G. Dwyer, and Daniel C. Isaksen. "Obstruction theory in model categories." Advances in Mathematics 181, no. 2 (January 2004): 396–416. http://dx.doi.org/10.1016/s0001-8708(03)00070-7.

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Elagin, Alexei D. "Descent theory for derived categories." Russian Mathematical Surveys 64, no. 4 (August 31, 2009): 748–49. http://dx.doi.org/10.1070/rm2009v064n04abeh004632.

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Galindo, César. "Clifford theory for tensor categories." Journal of the London Mathematical Society 83, no. 1 (November 28, 2010): 57–78. http://dx.doi.org/10.1112/jlms/jdq064.

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