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1
Mezroui, Yahya. "Projection orthogonale sur le graphe d’une relation linéaire fermé." Transactions of the American Mathematical Society 352, no. 6 (December 15, 1999): 2789–800. http://dx.doi.org/10.1090/s0002-9947-99-02410-1.
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Ledermann, Florian. "Classifying Cartographic Projections Based on Dynamic Analysis of Program Code." Abstracts of the ICA 2 (October 9, 2020): 1. http://dx.doi.org/10.5194/ica-abs-2-38-2020.
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Abstract. Analyzing a given map to identify its projection and other geometrical properties has long been an important aspect of cartographic analysis. If explicit information about the projection used in a particular map is not available, the properties of the cartographic transformation can sometimes be reconstructed from the map image. However, such a process of projection analysis requires significant manual labor and oversight.For digital maps, we usually expect the projection from geographic space to map space to have been calculated by a computer program. Such a program can be expected to contain the implementation of the mathematical rules of the projection and subsequent coordinate transformations such as translation and scaling. The program code, therefore, contains information that would allow an analyst to reliably identify map projections and other geometrical transformations applied to the input data.In the case of interactive online maps, the code generating the map is in fact delivered to the map user and could be used for cartographic analysis. The core idea of our novel method proposed for map analysis is to apply reverse engineering techniques on the code implementing the cartographic transformations in order to retrieve the properties of the applied map projection. However, automatic reasoning about computer code by way of static analysis (analyzing the source code without running it) is provably limited – for example, the code delivered to the map user may contain a whole library of different map projections, of which only a specific one may be actually used at runtime. Instead, we propose a dynamic analysis approach to observe and monitor the operations performed by the code as the program runs, and to retrieve the mathematical operations that have been used to calculate the coordinates of every graphical element on the map.The presented method produces, for every graphical element of the map, a transformation graph consisting of low-level mathematical operations. Cartographic projections can be identified as distinctive patterns in the transformation graph, and can be distinguished in a fully automatic way by matching a set of predefined patterns against a particular graph.Projections vary widely in their arithmetic structure, and therefore by the structure of the corresponding transformation graphs extracted from program code. Some projections can be computed directly using continuous equations involving trigonometric functions. Other projections involve solving nonlinear equations, which need to be solved by approximation. Composite projections use different projections depending on some threshold value. Yet other projections, such as the Robinson projection, define a table of predefined values, between which interpolation is used etc.. In each of these cases, we expect to find the operations corresponding to the mathematical structure of the projection in the transformation graph extracted by the presented method.For verifying the method, we have implemented the patterns of several well-known cartographic projections based on the literature and have used it on the transformation graphs extracted from a variety of sample programs. To ensure a diversity of implementations, we have evaluated programs using different and independent JavaScript implementations of projections, including the open source libraries D3.js, proj4js, Leaflet, OpenLayers, and informal implementations of example programs found online. For these case studies, we could successfully identify many projections based on identifying patterns in the transformation graph in a fully automated, unsupervised manner.In the future, the proposed method may be further developed for many innovative application scenarios, such as building a “cartographic search engine” or constructing novel tools for semi-automatic cartographic analysis and review.
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Dilworth, Stephen J., Denka Kutzarova, and Mikhail I. Ostrovskii. "Lipschitz-free Spaces on Finite Metric Spaces." Canadian Journal of Mathematics 72, no. 3 (February 13, 2019): 774–804. http://dx.doi.org/10.4153/s0008414x19000087.
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AbstractMain results of the paper are as follows:(1) For any finite metric space $M$ the Lipschitz-free space on $M$ contains a large well-complemented subspace that is close to $\ell _{1}^{n}$.(2) Lipschitz-free spaces on large classes of recursively defined sequences of graphs are not uniformly isomorphic to $\ell _{1}^{n}$ of the corresponding dimensions. These classes contain well-known families of diamond graphs and Laakso graphs.Interesting features of our approach are: (a) We consider averages over groups of cycle-preserving bijections of edge sets of graphs that are not necessarily graph automorphisms. (b) In the case of such recursive families of graphs as Laakso graphs, we use the well-known approach of Grünbaum (1960) and Rudin (1962) for estimating projection constants in the case where invariant projections are not unique.
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Bae, Yeolhui, Yugyeom Yi, Jeongmoo Lee, and Sungmo Kang. "Research on Definition of BLL Graphs of Knot Diagrams and its Applications." Korean Science Education Society for the Gifted 14, no. 3 (December 30, 2022): 229–36. http://dx.doi.org/10.29306/jseg.2022.14.3.229.
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This paper is the research on the Knot theory in Topology. A knot is a simple closed curve in ℝ and its projection onto a plane in ℝ is called a knot projection. As the results of this paper we define a BLL(Bidirectional Linear Link) graph for a knot projection which is a bidirectional linear link representing the relations between arcs of a knot projection and obtain some properties of the BLL graphs. We also define an Eulerian cycle of the BLL graph and an Eulerian cycle of a knot projection. As the main results of this paper, we obtain the equivalent conditions of being an alternation knot projection as follows: (1) an out-degree of every vertex of the corresponding BLL graph is 2; (2) the corresponding BLL graph has an Eulerian cycle; (3) the knot projection has an Eulerian cycle. As the subsequent study, using these results of the BLL graphs, we propose the analysis on the BLL graphs for deformation operation obtaining a new alternating knot projection, decision on the tricolorability of a knot projection, and a polynomial of a knot projection.
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Larose, Benoit. "Strongly Projective Graphs." Canadian Journal of Mathematics 54, no. 4 (August 1, 2002): 757–68. http://dx.doi.org/10.4153/cjm-2002-029-7.
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AbstractWe introduce the notion of strongly projective graph, and characterise these graphs in terms of their neighbourhood poset. We describe certain exponential graphs associated to complete graphs and odd cycles. We extend and generalise a result of Greenwell and Lovász [6]: if a connected graph G does not admit a homomorphism to K, where K is an odd cycle or a complete graph on at least 3 vertices, then the graph G × Ks admits, up to automorphisms of K, exactly s homomorphisms to K.
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Kaliszewski, S., Alex Kumjian, John Quigg, and Aidan Sims. "Topological realizations and fundamental groups of higher-rank graphs." Proceedings of the Edinburgh Mathematical Society 59, no. 1 (June 10, 2015): 143–68. http://dx.doi.org/10.1017/s0013091515000061.
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AbstractWe investigate topological realizations of higher-rank graphs. We show that the fundamental group of a higher-rank graph coincides with the fundamental group of its topological realization. We also show that topological realization of higher-rank graphs is a functor and that for each higher-rank graphΛ, this functor determines a category equivalence between the category of coverings ofΛand the category of coverings of its topological realization. We discuss how topological realization relates to two standard constructions fork-graphs: projective limits and crossed products by finitely generated free abelian groups.
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NIKKUNI, RYO. "COMPLETELY DISTINGUISHABLE PROJECTIONS OF SPATIAL GRAPHS." Journal of Knot Theory and Its Ramifications 15, no. 01 (January 2006): 11–19. http://dx.doi.org/10.1142/s0218216506004282.
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A generic immersion of a finite graph into the 2-space with p double points is said to be completely distinguishable if any two of the 2p embeddings of the graph into the 3-space obtained from the immersion by giving over/under information to each double point are not ambient isotopic in the 3-space. We show that only non-trivializable graphs and non-planar graphs have a non-trivial completely distinguishable immersion. We give examples of non-trivial completely distinguishable immersions of several non-trivializable graphs, the complete graph on n vertices and the complete bipartite graph on m + n vertices.
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HUH, YOUNGSIK. "AN ELEMENTARY SET FOR EMBEDDED BOUQUET GRAPHS WITH TWO CYCLES." Journal of Knot Theory and Its Ramifications 20, no. 02 (February 2011): 305–25. http://dx.doi.org/10.1142/s0218216511008796.
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A projection of the bouquet graph B with two cycles is said to be trivial if only trivial embeddings are obtained from the projection. In this paper a finite set of nontrivial embeddings of B is shown to be minimal among those which produce all nontrivial projections of B.
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Filippova, P. A. "Values of the weight system on a family of graphs that are not the intersection graphs of chord diagrams." Sbornik: Mathematics 213, no. 2 (February 1, 2022): 235–67. http://dx.doi.org/10.1070/sm9519.
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Abstract The Chmutov-Lando theorem claims that the value of a weight system (a function on the chord diagrams that satisfies the four-term Vassiliev relations) corresponding to the Lie algebra depends only on the intersection graph of the chord diagram. We compute the values of the weight system at the graphs in several infinite series, which are the joins of a graph with a small number of vertices and a discrete graph. In particular, we calculate these values for a series in which the initial graph is the cycle on five vertices; the graphs in this series, apart from the initial one, are not intersection graphs. We also derive a formula for the generating functions of the projections of graphs equal to the joins of an arbitrary graph and a discrete graph to the subspace of primitive elements of the Hopf algebra of graphs. Using the formula thus obtained, we calculate the values of the weight system at projections of the graphs of the indicated form onto the subspace of primitive elements. Our calculations confirm Lando’s conjecture concerning the values of the weight system at projections onto the subspace of primitives. Bibliography: 17 titles.
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HUH, YOUNGSIK, and KOUKI TANIYAMA. "IDENTIFIABLE PROJECTIONS OF SPATIAL GRAPHS." Journal of Knot Theory and Its Ramifications 13, no. 08 (December 2004): 991–98. http://dx.doi.org/10.1142/s0218216504003640.
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A generic map from a finite graph to the 2-space is called identifiable if any two embeddings of the graph into the 3-space obtained by lifting the map with respect to the natural projection from the 3-space to the 2-space are ambient isotopic in the 3-space. We show that only planar graphs have identifiable maps. We characterize the identifiable maps for some planar graphs.
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NIKKUNI, RYO, MAKOTO OZAWA, KOUKI TANIYAMA, and YUKIHIRO TSUTSUMI. "NEWLY FOUND FORBIDDEN GRAPHS FOR TRIVIALIZABILITY." Journal of Knot Theory and Its Ramifications 14, no. 04 (June 2005): 523–38. http://dx.doi.org/10.1142/s0218216505003932.
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A planar graph is said to be trivializable if every regular projection of the graph produces a trivial spatial embedding by giving some over/under informations to the double points. Every minor of a trivializable graph is also trivializable, thus the set of forbidden graphs is finite. Seven forbidden graphs for the trivializability were previously known. In this paper, we exhibit nine more forbidden graphs.
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Hjouj, Fawaz. "On Tomography with Unknown Orientation." Journal of Mathematical Sciences & Computer Applications 2, no. 2 (June 10, 2017): 125–35. http://dx.doi.org/10.5147/jmsca.v2i1.99.
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We consider the two-dimensional parallel beam Tomography problem in which both the object being imaged and the projection directions are unknown. The angles of projections need not to be uniformly distributed. Our solution combines two known approaches: the Geometric Moment and the Graph Laplacian. After sorting the projections using the Graph Laplacian method we create a one to one moment function of the angles. We then solve for each angle uniquely.
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CAELLI, TERRY, and SERHIY KOSINOV. "INEXACT GRAPH MATCHING USING EIGEN-SUBSPACE PROJECTION CLUSTERING." International Journal of Pattern Recognition and Artificial Intelligence 18, no. 03 (May 2004): 329–54. http://dx.doi.org/10.1142/s0218001404003186.
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Graph eigenspaces have been used to encode many different properties of graphs. In this paper we explore how such methods can be used for solving inexact graph matching (the matching of sets of vertices in one graph to those in another) having the same or different numbers of vertices. In this case we explore eigen-subspace projections and vertex clustering (EPS) methods. The correspondence algorithm enables the EPC method to discover a range of correspondence relationships from one-to-one vertex matching to that of inexact (many-to-many) matching of structurally similar subgraphs based on the similarities of their vertex connectivities defined by their positions in the common subspace. Examples in shape recognition and random graphs are used to illustrate this method.
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Sumet Mehta. "Generalized Multi-manifold Graph Ensemble Embedding for Multi-View Dimensionality Reduction." Lahore Garrison University Research Journal of Computer Science and Information Technology 4, no. 4 (December 28, 2020): 55–72. http://dx.doi.org/10.54692/lgurjcsit.2020.0404109.
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In this paper, we propose a new dimension reduction (DR) algorithm called ensemble graph-based locality preserving projections (EGLPP); to overcome the neighborhood size k sensitivity in locally preserving projections (LPP). EGLPP constructs a homogeneous ensemble of adjacency graphs by varying neighborhood size k and finally uses the integrated embedded graph to optimize the low-dimensional projections. Furthermore, to appropriately handle the intrinsic geometrical structure of the multi-view data and overcome the dimensionality curse, we propose a generalized multi-manifold graph ensemble embedding framework (MLGEE). MLGEE aims to utilize multi-manifold graphs for the adjacency estimation with automatically weight each manifold to derive the integrated heterogeneous graph. Experimental results on various computer vision databases verify the effectiveness of proposed EGLPP and MLGEE over existing comparative DR methods.
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HANAKI, RYO. "REGULAR PROJECTIONS OF KNOTTED DOUBLE-HANDCUFF GRAPHS." Journal of Knot Theory and Its Ramifications 18, no. 11 (November 2009): 1475–92. http://dx.doi.org/10.1142/s0218216509007567.
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A finite set of specific knotted double-handcuff graphs is shown to be minimal among those which produce all projections of knotted double-handcuff graphs. In addition, we show that a double-handcuff graph has no strongly almost trivial spatial embeddings.
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Aceves, Elaina. "A study of projections of 2-bouquet graphs." Journal of Knot Theory and Its Ramifications 26, no. 05 (April 2017): 1750025. http://dx.doi.org/10.1142/s0218216517500250.
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We extend the concepts of trivializing and knotting numbers for knots to spatial graphs and 2-bouquet graphs, in particular. Furthermore, we calculate the trivializing and knotting numbers for projections and pseudodiagrams of 2-bouquet spatial graphs based on the number of precrossings and the placement of the precrossings in the pseudodiagram of the spatial graph.
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Petrenjuk, Volodymyr, Dmytro Petreniuk, and Oleh Oryshaka. "Structure of Projective Planar Subgraphs of the Graph Obstructions for Fixed Surface." Cybernetics and Computer Technologies, no. 2 (September 30, 2022): 13–30. http://dx.doi.org/10.34229/2707-451x.22.2.2.
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Consider the problem of studying the metric properties of a subgraph G\v, where v is an arbitrary vertex of obstruction graphs G of a nonorientable genus, which will determine the sets of points of attachment of one subgraph to another and allow constructing prototypes of graphs-obstruction with number of vertices greater than 10 nonorientable genus greater than 1. This problem is related to Erdosh's hypothesis [3] on the coverage of obstruction graphs of an undirected surface of the genus k, where k > 0, the smallest inclusion of the set of k + 1st graph of the homeomorphic K5, or K3,3, in [4] constructively proved for 35 minors of obstruction graphs of the projective plane, a set of 62 with no more than 10 vertices of obstruction graphs and their splits for the Klein surface, as well as some obstruction graphs for other surfaces. In [5], the existence of a finite set of obstruction graphs for a non-orienting surface was proved. A similar problem was considered in [6], where models or prototypes of obstruction graphs were considered. The prototype of the graph-obstruction of the undirected genus, we will call the graphs that have their own subgraph graph-obstruction of the undirected genus. In [7, 8] the tangent problem of covering the set of vertices with the smallest number of cycles-boundaries of 2-cells was considered, the concept of cell distance is given in [9, 10], where the boundaries of an oriented genus of graphs formed from planar graphs and a simple star glued to some of its peaks. Hypothetically, it is possible to obtain them by recursive φ-transformation of the graph-obstruction of the projective plane and a copy of its planar subgraph given on vertices, edges or parts of edges, or simple chains, i.e. achievable parts of the so-called graph-basis (graph of homeomorphic graph Kuratovsky plane). We assume that instead of one subgraph there can be several copies of subgraphs of graphs-obstructions of the projective plane.The article has an introduction and two parts, in which the structural properties of subgraphs of obstruction graphs for an undirected surface, presented as a φ-image of one of the Kuratovsky graphs and at least one planar graph, are investigated. The metric properties of the minimal embeddings of the subgraphs of the obstruction graphs for undirected surfaces are considered, and the main result is Theorems 1, 2, and Lemma 3 as the basis of the prototype construction algorithm. Keywords: φ-transformation of graphs, nonorientable surface, prototypes of graph-obstruction.
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Fukś, Henryk, Babak Farzad, and Yi Cao. "A model of language inflection graphs." International Journal of Modern Physics C 25, no. 06 (April 23, 2014): 1450013. http://dx.doi.org/10.1142/s0129183114500132.
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Inflection graphs are highly complex networks representing relationships between inflectional forms of words in human languages. For so-called synthetic languages, such as Latin or Polish, they have particularly interesting structure due to the abundance of inflectional forms. We construct the simplest form of inflection graphs, namely a bipartite graph in which one group of vertices corresponds to dictionary headwords and the other group to inflected forms encountered in a given text. We, then, study projection of this graph on the set of headwords. The projection decomposes into a large number of connected components, to be called word groups. Distribution of sizes of word group exhibits some remarkable properties, resembling cluster distribution in a lattice percolation near the critical point. We propose a simple model which produces graphs of this type, reproducing the desired component distribution and other topological features.
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Douar, Brahim, Chiraz Latiri, Michel Liquiere, and Yahya Slimani. "A Projection Bias in Frequent Subgraph Mining Can Make a Difference." International Journal on Artificial Intelligence Tools 23, no. 05 (October 2014): 1450005. http://dx.doi.org/10.1142/s0218213014500055.
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The aim of the frequent subgraph mining task is to find frequently occurring subgraphs in a large graph database. However, this task is a thriving challenge, as graph and subgraph isomorphisms play a key role throughout the computations. Since subgraph isomorphism testing is a hard problem, subgraph miners are exponential in runtime. To alleviate the complexity issue, we propose to introduce a bias in the projection operator and instead of using the costly subgraph isomorphism projection, one can use a polynomial projection having a semantically-valid structural interpretation. This paper presents a new projection operator for graphs named AC-projection, which exhibits nice theoretical complexity properties. We study the size of the search space as well as some practical properties of the projection operator. We also introduce a novel breadth-first algorithm for frequent AC-reduced subgraphs mining. Then, we prove experimentally that we can achieve an important performance gain (polynomial complexity projection) without or with non-significant loss of discovered patterns in terms of quality.
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Zhang, Xianhang, Hanchen Wang, Jianke Yu, Chen Chen, Xiaoyang Wang, and Wenjie Zhang. "Polarity-based graph neural network for sign prediction in signed bipartite graphs." World Wide Web 25, no. 2 (February 16, 2022): 471–87. http://dx.doi.org/10.1007/s11280-022-01015-4.
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AbstractAs a fundamental data structure, graphs are ubiquitous in various applications. Among all types of graphs, signed bipartite graphs contain complex structures with positive and negative links as well as bipartite settings, on which conventional graph analysis algorithms are no longer applicable. Previous works mainly focus on unipartite signed graphs or unsigned bipartite graphs separately. Several models are proposed for applications on the signed bipartite graphs by utilizing the heuristic structural information. However, these methods have limited capability to fully capture the information hidden in such graphs. In this paper, we propose the first graph neural network on signed bipartite graphs, namely Polarity-based Graph Convolutional Network (PbGCN), for sign prediction task with the help of balance theory. We introduce the novel polarity attribute to signed bipartite graphs, based on which we construct one-mode projection graphs to allow the GNNs to aggregate information between the same type nodes. Extensive experiments on five datasets demonstrate the effectiveness of our proposed techniques.
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Sawollek, Jörg. "Embeddings of 4-Regular Graphs into 3-Space." Journal of Knot Theory and Its Ramifications 06, no. 05 (October 1997): 727–49. http://dx.doi.org/10.1142/s0218216597000406.
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Embeddings of 4-regular graphs into 3-space are examined by studying graph diagrams, i.e. projections of embedded graphs to an appropriate plane. New diagrams can be constructed from the old ones by replacing graph vertices with rational tangles, and these diagrams lead to topological invariants of embedded graphs. The new invariants are calculated for some examples, in particular for classes of alternating diagrams of the figure-eight graph. As an application, it is shown that these diagrams have minimal crossing number, which gives generalizations to some of the so-called Tait conjectures.
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SHTYLLA, BLERTA, and LOUIS ZULLI. "AN EXTENSION OF THE JONES POLYNOMIAL OF CLASSICAL KNOTS." Journal of Knot Theory and Its Ramifications 15, no. 01 (January 2006): 81–100. http://dx.doi.org/10.1142/s0218216506004294.
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We define a linear algebraic extension of the Jones polynomial of classical knots, and prove that certain key properties of the classical Jones polynomial are properties of the extension. This shows that these properties are linear algebraic in nature, not topological. We identify a topological property of the classical Jones polynomial, that is, a property of the classical Jones polynomial that the extension does not possess. We discuss ortho-projection matrices, ortho-projection graphs, and their Jones polynomials. We classify, up to isomorphism, the connected ortho-projection graphs with at most eight vertices, and show that each such isomorphism class corresponds to a prime alternating classical knot diagram. We give an example of a connected ortho-projection graph with nine vertices that does not correspond to such a diagram.
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Arockiamary, S. Teresa, C. Meera, and V. Santhi. "Connectedness in Projection Graphs of Commutative Rings." Indian Journal Of Science And Technology 16, SP3 (December 15, 2023): 55–63. http://dx.doi.org/10.17485/ijst/v16isp3.icrtam293.
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Shen, Xiang-Jun, Stanley Ebhohimhen Abhadiomhen, Yang Yang, Zhifeng Liu, and Sirui Tian. "Edge Structure Learning via Low Rank Residuals for Robust Image Classification." Proceedings of the AAAI Conference on Artificial Intelligence 37, no. 2 (June 26, 2023): 2236–44. http://dx.doi.org/10.1609/aaai.v37i2.25318.
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Traditional low-rank methods overlook residuals as corruptions, but we discovered that low-rank residuals actually keep image edges together with corrupt components. Therefore, filtering out such structural information could hamper the discriminative details in images, especially in heavy corruptions. In order to address this limitation, this paper proposes a novel method named ESL-LRR, which preserves image edges by finding image projections from low-rank residuals. Specifically, our approach is built in a manifold learning framework where residuals are regarded as another view of image data. Edge preserved image projections are then pursued using a dynamic affinity graph regularization to capture the more accurate similarity between residuals while suppressing the influence of corrupt ones. With this adaptive approach, the proposed method can also find image intrinsic low-rank representation, and much discriminative edge preserved projections. As a result, a new classification strategy is introduced, aligning both modalities to enhance accuracy. Experiments are conducted on several benchmark image datasets, including MNIST, LFW, and COIL100. The results show that the proposed method has clear advantages over compared state-of-the-art (SOTA) methods, such as Low-Rank Embedding (LRE), Low-Rank Preserving Projection via Graph Regularized Reconstruction (LRPP_GRR), and Feature Selective Projection (FSP) with more than 2% improvement, particularly in corrupted cases.
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Bach, Nguyen Gia, Chanh Minh Tran, Tho Nguyen Duc, Phan Xuan Tan, and Eiji Kamioka. "Novel Projection Schemes for Graph-Based Light Field Coding." Sensors 22, no. 13 (June 30, 2022): 4948. http://dx.doi.org/10.3390/s22134948.
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In light field compression, graph-based coding is powerful to exploit signal redundancy along irregular shapes and obtains good energy compaction. However, apart from high time complexity to process high dimensional graphs, their graph construction method is highly sensitive to the accuracy of disparity information between viewpoints. In real-world light field or synthetic light field generated by computer software, the use of disparity information for super-rays projection might suffer from inaccuracy due to vignetting effect and large disparity between views in the two types of light fields, respectively. This paper introduces two novel projection schemes resulting in less error in disparity information, in which one projection scheme can also significantly reduce computation time for both encoder and decoder. Experimental results show projection quality of super-pixels across views can be considerably enhanced using the proposals, along with rate-distortion performance when compared against original projection scheme and HEVC-based or JPEG Pleno-based coding approaches.
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ZHOU, JIN-XIN, and YAN-QUAN FENG. "TETRAVALENT s-TRANSITIVE GRAPHS OF ORDER TWICE A PRIME POWER." Journal of the Australian Mathematical Society 88, no. 2 (April 2010): 277–88. http://dx.doi.org/10.1017/s1446788710000066.
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AbstractA graph is s-transitive if its automorphism group acts transitively on s-arcs but not on (s+1)-arcs in the graph. Let X be a connected tetravalent s-transitive graph of order twice a prime power. In this paper it is shown that s=1,2,3 or 4. Furthermore, if s=2, then X is a normal cover of one of the following graphs: the 4-cube, the complete graph of order 5, the complete bipartite graph K5,5 minus a 1-factor, or K7,7 minus a point-hyperplane incidence graph of the three-dimensional projective geometry PG(2,2); if s=3, then X is a normal cover of the complete bipartite graph of order 4; if s=4, then X is a normal cover of the point-hyperplane incidence graph of the three-dimensional projective geometry PG(2,3). As an application, we classify the tetravalent s-transitive graphs of order 2p2 for prime p.
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VARADI, Zoltan. "How Graphs can Improve Targeting of Employee Trainings?" Eurasia Proceedings of Educational and Social Sciences 31 (October 30, 2023): 135–42. http://dx.doi.org/10.55549/epess.1381972.
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Abstract: Skills matrices, also known as competency matrices, can help shift managers in a manufacturing environment in proper allocation of operators to workplaces. However, improving the skills portfolio of shift workers is often based on perceived problems with skills of absent workers. This study examines a company’s skills matrices of the 3 shifts, questioning if graph metrics can help estimating substitutability, i.e., robustness of skills portfolio of workers to absenteeism; and how can graph mapping help better targeting trainings. The author has constructed bipartite graphs where one set of nodes are from the set of competencies and the other set of nodes are from the set of workers; and evaluated metrics comparing the skills portfolio of each shift. One projection of the bipartite graph shows the interlinks between people: when two workers are connected, they share the same skill and can substitute each other. The overall level of substitutability of people is then measured with the average degree of nodes of the projection graph. Weak connectedness, that is, low k values can highlight risks and exposedness to fallout of the respective workers. Disjoint graphs indicate if there is an option for a sub-team setup based on competencies. The other projection has an edge between to skills if and only if there is minimum one worker who is capable for both. Disjoint subgraphs of skills are helping team formation based on competencies.
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Petrenjuk, Volodymyr, and Dmytro Petreniuk. "Models of Klein Surface Obstruction Graphs." Cybernetics and Computer Technologies, no. 1 (March 29, 2024): 47–63. http://dx.doi.org/10.34229/2707-451x.24.1.4.
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The task of researching the structure of graphs of given connectivity, which are obstructions for a given surface of non-oriented kind, and building their models, from which obstruction graphs are formed by removing or compressing a set of edges, is considered. The issue of edge coverage of an obstruction graph of a given kind with a minimum number of quasi-stars with centers – planar graphs that have given sets of points and all edges are significant with respect to the reachability number 2 on the Euclidean plane and has reachability on the projective plane or Klein surface, is considered. K4, K2,3 or a degenerate graph. The task of researching the structure of graphs of undirected kind was considered [4–6]. In [7], the set of minors for the projective plane was compressed to 12 basic minors using the method of relative components, and a set of 62 minors of the Klein surface was constructed. To do this, we considered all non-isomorphic minimal embeddings of each of the basic minors and found the set of all different pairs of vertices that are reachable on the projective plane during the operations of removing or compressing an arbitrary edge of this graph to a point, then a pair of non-adjacent graph vertices was attached to the selected pair of points. In [8], the number of 2-connected obstruction graphs for the Klein surface was calculated, part of the diagrams of these graphs is given in [10]. Note that the following definition of the cell distance is similar to that in [11].Our approach, as a continuation of [9], will consist in finding the edge covering of an obstruction graph of a given kind by the minimum number of subgraphs of the covering from the number of quasi-stars with centers - graphs with essential edges relative to the number of reachability or nonorientable genus during compression to a point or removal operations edges relative to a given set of points with reachability number 2 relative to the Euclidean plane and reachable on projective planes or Klein surfaces, for example, these are subsets of the set of points of graphs K4, K2,3, K5\e, Kr, r >= 2, or graph-obstructions of the projective plane. We also found the necessary conditions for constructing obstruction graphs for the Klein surface by identifying pairs of center points and hanging vertices of three quasi-stars, thus we have the basis of an algorithm for constructing a larger number of obstruction graphs for the Klein surface. Hypothetically, a graph-obstruction of a given nonorientable genus has the form of a cylindrical surface with n, n >= 2, disks-bases and a side part, which can have common sets of points on the boundaries and on which are embedded, at least in part, the graph-centers of quasi-stars having a given set of reachability points 2 on the Euclidean plane, and on the side surface there are hanging edges that intersect on the plane and are inserted without crossing with the help of Mobius strips glued to the side surface. At the same time, the edges will have at least two nesting options in the side part of the cylindrical surface, but no more than the number of glued Mobius strips, thanks to which each hanging edge will nest on the Mobius strip, either with only one edge or with two adjacent edges. We have found the necessary conditions for constructing models of obstruction graphs for the Klein surface by identifying pairs of centers and hanging vertices of three quasi-stars, thus we have the basis of an algorithm for constructing a larger number of obstruction graphs for the Klein surface. The main result: statements 1, 2, 3 and the algorithm for constructing models of 3-connected graph-obstructions of the Klein surface. Keywords: φ-transformation of graphs, nonorientable surface, prototypes of graph-obstruction.
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Petrenjuk, V. I., and D. A. Petrenjuk. "About Structure of Graph Obstructions for Klein Surface with 9 Vertices." Cybernetics and Computer Technologies, no. 4 (December 31, 2020): 65–86. http://dx.doi.org/10.34229/2707-451x.20.4.5.
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The structure of the 9 vertex obstructive graphs for the nonorientable surface of the genus 2 is established by the method of j-transformations of the graphs. The problem of establishing the structural properties of 9 vertex obstruction graphs for the surface of the undirected genus 2 by the method of j-transformation of graphs is considered. The article has an introduction and 5 sections. The introduction contains the main definitions, which are illustrated, to some extent, in Section 1, which provides several statements about their properties. Sections 2 – 4 investigate the structural properties of 9 vertex obstruction graphs for an undirected surface by presenting as a j-image of several graphs homeomorphic to one of the Kuratovsky graphs and at least one planar or projective-planar graph. Section 5 contains a new version of the proof of the statement about the peculiarities of the minimal embeddings of finite graphs in nonorientable surfaces, namely, that, in contrast to oriented surfaces, cell boundaries do not contain repeated edges. Also in section 5 the other properties peculiar to embeddings of graphs to non-oriented surfaces and the main result are given. The main result is Theorem 1. Each obstruction graph H for a non-oriented surface N2 of genus 2 satisfies the following. 1. An arbitrary edge u,u = (a,b) is placed on the Mebius strip by some minimal embedding of the graph H in N3 and there exists a locally projective-planar subgraph K of the graph H \ u which satisfies the condition: (tK({a,b},N3)=1)˄(tK\u({a,b},N2)=2), where tK({a,b},N) is the number of reachability of the set {a,b} on the nonorientable surface N; 2. There exists the smallest inclusion of many different subgraphs Ki of a 2-connected graph H homeomorphic to the graph K+e, where K is a locally planar subgraph of the graph H (at least K+e is homemorphic to K5 or K3,3), which covers the set of edges of the graph H. Keywords: graph, Klein surface, graph structure, graph obstruction, non-oriented surface, Möbius strip.
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Dogan, I., and A. Akpinar. "On Distance in Some Finite Planes and Graphs Arising from Those Planes." Journal of Mathematics 2021 (January 28, 2021): 1–17. http://dx.doi.org/10.1155/2021/6668682.
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In this paper, affine and projective graphs are obtained from affine and projective planes of order p r by accepting a line as a path. Some properties of these affine and projective graphs are investigated. Moreover, a definition of distance is given in the affine and projective planes of order p r and, with the help of this distance definition, the point or points having the most advantageous (central) position in the corresponding graphs are determined, with some examples being given. In addition, the concepts of a circle, ellipse, hyperbola, and parabola, which are well known for the Euclidean plane, are carried over to these finite planes. Finally, the roles of finite affine and projective Klingenberg planes in all the results obtained are considered and their equivalences in graph applications are discussed.
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KHEIRABADI, M., and A. R. MOGHADDAMFAR. "RECOGNIZING SOME FINITE SIMPLE GROUPS BY NONCOMMUTING GRAPH." Journal of Algebra and Its Applications 11, no. 04 (July 31, 2012): 1250077. http://dx.doi.org/10.1142/s0219498812500776.
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Let G be a nonabelian group. We define the noncommuting graph ∇(G) of G as follows: its vertex set is G\Z(G), the noncentral elements of G, and two distinct vertices x and y of ∇(G) are joined by an edge if and only if x and y do not commute as elements of G, i.e. [x, y] ≠ 1. The finite group L is said to be recognizable by noncommuting graph if, for every finite group G, ∇(G) ≅ ∇ (L) implies G ≅ L. In the present article, it is shown that the noncommuting graph of a group with trivial center can determine its prime graph. From this, the following theorem is derived. If two finite groups with trivial centers have isomorphic noncommuting graphs, then their prime graphs coincide. It is also proved that the projective special unitary groups U4(4), U4(8), U4(9), U4(11), U4(13), U4(16), U4(17) and the projective special linear groups L9(2), L16(2) are recognizable by noncommuting graph.
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Hogben, Leslie, Kevin Palmowski, David Roberson, and Simone Severini. "Orthogonal Representations, Projective Rank, and Fractional Minimum Positive Semidefinite Rank: Connections and New Directions." Electronic Journal of Linear Algebra 32 (February 6, 2017): 98–115. http://dx.doi.org/10.13001/1081-3810.3102.
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Fractional minimum positive semidefinite rank is defined from r-fold faithful orthogonal representations and it is shown that the projective rank of any graph equals the fractional minimum positive semidefinite rank of its complement. An r-fold version of the traditional definition of minimum positive semidefinite rank of a graph using Hermitian matrices that fit the graph is also presented. This paper also introduces r-fold orthogonal representations of graphs and formalizes the understanding of projective rank as fractional orthogonal rank. Connections of these concepts to quantum theory, including Tsirelson's problem, are discussed.
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FENG, GUIYU, DAVID ZHANG, JIAN YANG, and DEWEN HU. "A THEORETICAL FRAMEWORK FOR MATRIX-BASED FEATURE EXTRACTION ALGORITHMS WITH ITS APPLICATION TO IMAGE RECOGNITION." International Journal of Image and Graphics 08, no. 01 (January 2008): 1–23. http://dx.doi.org/10.1142/s0219467808002940.
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Recently proposed matrix-based methods, two-dimensional Principal Component Analysis (2DPCA), two-dimensional Linear Discriminant Analysis (2DLDA) and two-dimensional Locality Preserving Projections (2DLPP) have been shown to be effective ways to avoid the problems of high dimensionality and small sample sizes that are associated with vector-based methods. In this paper, we propose a general theoretical framework for matrix-based feature extraction algorithms from the point of view of graph embedding. Our framework can be applied to extend two recently proposed vector-based algorithms, i.e. Unsupervised Discriminant Projection (UDP) and Marginal Fisher Analysis (MFA) algorithms, to their matrix-based versions. Further, our framework can also be used as a platform to generate new matrix-based feature extraction algorithms by designing meaningful graphs, e.g. two-dimensional Discriminant Embedding Analysis (2DDEA) in this paper. It is shown that 2DLDA is actually a special case of the 2DDEA method. Experiments on three publicly available image databases demonstrate the effectiveness of the proposed algorithm. Our results fit into the scene for a better picture about the matrix-based feature extraction algorithms.
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Prakash, Sudhir, Rakesh Kumar, Piyush Rai, Shiksha Jain, Manish Singh, Rajnish Pandey, Saanidhya Dubey, Shobhit Srivatava, and Anoop K. Srivastava. "A Study of Chemical Compound of Graph with help of Computer Coding." Turkish Journal of Computer and Mathematics Education (TURCOMAT) 10, no. 3 (December 13, 2019): 1553–64. http://dx.doi.org/10.61841/turcomat.v10i3.14357.
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In this research work we study about the outline of graph structures algorithms required for the recognition of chemical graphs by a computer automatic encoding of organic chemical structures into the line formula notation. During the study we will first classify chemical graph then find a planar projection and fundamental cycles of a chemical graph with its characterization. Then we draw that for automatic construction code from the connection table for this we need some algorithms. Thus a graph structure language like GRAAL should prove useful for chemical coding systems. Further development along these lines should be helpful in coding knotted chemical structures.
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Pan, Lei, Hengchao Li, Xiang Dai, Ying Cui, Xifeng Huang, and Lican Dai. "Latent Low-Rank Projection Learning with Graph Regularization for Feature Extraction of Hyperspectral Images." Remote Sensing 14, no. 13 (June 27, 2022): 3078. http://dx.doi.org/10.3390/rs14133078.
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Due to the great benefit of rich spectral information, hyperspectral images (HSIs) have been successfully applied in many fields. However, some problems of concern also limit their further applications, such as high dimension and expensive labeling. To address these issues, an unsupervised latent low-rank projection learning with graph regularization (LatLRPL) method is presented for feature extraction and classification of HSIs in this paper, in which discriminative features can be extracted from the view of latent space by decomposing the latent low-rank matrix into two different matrices, also benefiting from the preservation of intrinsic subspace structures by the graph regularization. Different from the graph embedding-based methods that need two phases to obtain the low-dimensional projections, one step is enough for LatLRPL by constructing the integrated projection learning model, reducing the complexity and simultaneously improving the robustness. To improve the performance, a simple but effective strategy is exploited by conducting the local weighted average on the pixels in a sliding window for HSIs. Experiments on the Indian Pines and Pavia University datasets demonstrate the superiority of the proposed LatLRPL method.
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Ivanov, Alexander A. "The non-existence of a super-Janko group." Glasnik Matematicki 58, no. 2 (December 27, 2023): 289–305. http://dx.doi.org/10.3336/gm.58.2.09.
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Locally projective graphs in Mathieu–Conway–Monster series appear in thin–thick pairs. A possible thick extension of a thin locally projective graph associated with the fourth Janko group has been questioned for a while. Such an extension could lead, if not to a new sporadic simple group, to something equally exciting. This paper resolves this issue ultimately in the non-existence form confirming that the list of 26 sporadic simple groups, although mysterious, is now stable. The result in fact concludes the classification project of locally projective graphs, which has been running for some twenty years.
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Nayyeri, Mojtaba, Sahar Vahdati, Can Aykul, and Jens Lehmann. "5* Knowledge Graph Embeddings with Projective Transformations." Proceedings of the AAAI Conference on Artificial Intelligence 35, no. 10 (May 18, 2021): 9064–72. http://dx.doi.org/10.1609/aaai.v35i10.17095.
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Performing link prediction using knowledge graph embedding models has become a popular approach for knowledge graph completion. Such models employ a transformation function that maps nodes via edges into a vector space in order to measure the likelihood of the links. While mapping the individual nodes, the structure of subgraphs is also transformed. Most of the embedding models designed in Euclidean geometry usually support a single transformation type -- often translation or rotation, which is suitable for learning on graphs with small differences in neighboring subgraphs. However, multi-relational knowledge graphs often include multiple subgraph structures in a neighborhood (e.g.~combinations of path and loop structures), which current embedding models do not capture well. To tackle this problem, we propose a novel KGE model 5*E in projective geometry, which supports multiple simultaneous transformations -- specifically inversion, reflection, translation, rotation, and homothety. The model has several favorable theoretical properties and subsumes the existing approaches. It outperforms them on most widely used link prediction benchmarks
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Janson, Svante. "The Numbers of Spanning Trees, Hamilton Cycles and Perfect Matchings in a Random Graph." Combinatorics, Probability and Computing 3, no. 1 (March 1994): 97–126. http://dx.doi.org/10.1017/s0963548300001012.
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The numbers of spanning trees, Hamilton cycles and perfect matchings in a random graph Gnm are shown to be asymptotically normal if m is neither too large nor too small. At the lowest limit m ≍ n3/2, these numbers are asymptotically log-normal. For Gnp, the numbers are asymptotically log-normal for a wide range of p, including p constant. The same results are obtained for random directed graphs and bipartite graphs. The results are proved using decomposition and projection methods.
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Li, Haohao, Mingliang Gao, Huibing Wang, and Gwanggil Jeon. "Multi-View Projection Learning via Adaptive Graph Embedding for Dimensionality Reduction." Electronics 12, no. 13 (July 3, 2023): 2934. http://dx.doi.org/10.3390/electronics12132934.
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In order to explore complex structures and relationships hidden in data, plenty of graph-based dimensionality reduction methods have been widely investigated and extended to the multi-view learning field. For multi-view dimensionality reduction, the key point is extracting the complementary and compatible multi-view information to analyze the complex underlying structure of the samples, which is still a challenging task. We propose a novel multi-view dimensionality reduction algorithm that integrates underlying structure learning and dimensionality reduction for each view into one framework. Because the prespecified graph derived from original noisy high-dimensional data is usually low-quality, the subspace constructed based on such a graph is also low-quality. To obtain the optimal graph for dimensionality reduction, we propose a framework that learns the affinity based on the low-dimensional representation of all views and performs the dimensionality reduction based on it jointly. Although original data is noisy, the local structure information of them is also valuable. Therefore, in the graph learning process, we also introduce the information of predefined graphs based on each view feature into the optimal graph. Moreover, assigning the weight to each view based on its importance is essential in multi-view learning, the proposed GoMPL automatically allocates an appropriate weight to each view in the graph learning process. The obtained optimal graph is then adopted to learn the projection matrix for each individual view by graph embedding. We provide an effective alternate update method for learning the optimal graph and optimal subspace jointly for each view. We conduct many experiments on various benchmark datasets to evaluate the effectiveness of the proposed method.
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DAMIAN, MIRELA, and KRISTIN RAUDONIS. "YAO GRAPHS SPAN THETA GRAPHS." Discrete Mathematics, Algorithms and Applications 04, no. 02 (June 2012): 1250024. http://dx.doi.org/10.1142/s1793830912500243.
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Yao and Theta graphs are defined for a given point set and a fixed integer k > 0. The space around each point is divided into k cones of equal angle, and each point is connected to a nearest neighbor in each cone. The difference between Yao and Theta graphs is in the way the nearest neighbor is defined: Yao graphs minimize the Euclidean distance between a point and its neighbor, and Theta graphs minimize the Euclidean distance between a point and the orthogonal projection of its neighbor on the bisector of the hosting cone. We prove that, corresponding to each edge of the Theta graph Θ6, there is a path in the Yao graph Y6 whose length is at most 8.82 times the edge length. Combined with the result of Bonichon et al., who prove an upper bound of 2 on the stretch factor of Θ6, we obtain an upper bound of 17.64 on the stretch factor of Y6.
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Thrun, Michael C., Julian Märte, and Quirin Stier. "Analyzing Quality Measurements for Dimensionality Reduction." Machine Learning and Knowledge Extraction 5, no. 3 (August 21, 2023): 1076–118. http://dx.doi.org/10.3390/make5030056.
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Dimensionality reduction methods can be used to project high-dimensional data into low-dimensional space. If the output space is restricted to two dimensions, the result is a scatter plot whose goal is to present insightful visualizations of distance- and density-based structures. The topological invariance of dimension indicates that the two-dimensional similarities in the scatter plot cannot coercively represent high-dimensional distances. In praxis, projections of several datasets with distance- and density-based structures show a misleading interpretation of the underlying structures. The examples outline that the evaluation of projections remains essential. Here, 19 unsupervised quality measurements (QM) are grouped into semantic classes with the aid of graph theory. We use three representative benchmark datasets to show that QMs fail to evaluate the projections of straightforward structures when common methods such as Principal Component Analysis (PCA), Uniform Manifold Approximation projection, or t-distributed stochastic neighbor embedding (t-SNE) are applied. This work shows that unsupervised QMs are biased towards assumed underlying structures. Based on insights gained from graph theory, we propose a new quality measurement called the Gabriel Classification Error (GCE). This work demonstrates that GCE can make an unbiased evaluation of projections. The GCE is accessible within the R package DR quality available on CRAN.
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Chen, Zhaoliang, Zhihao Wu, Shiping Wang, and Wenzhong Guo. "Dual Low-Rank Graph Autoencoder for Semantic and Topological Networks." Proceedings of the AAAI Conference on Artificial Intelligence 37, no. 4 (June 26, 2023): 4191–98. http://dx.doi.org/10.1609/aaai.v37i4.25536.
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Due to the powerful capability to gather the information of neighborhood nodes, Graph Convolutional Network (GCN) has become a widely explored hotspot in recent years. As a well-established extension, Graph AutoEncoder (GAE) succeeds in mining underlying node representations via evaluating the quality of adjacency matrix reconstruction from learned features. However, limited works on GAE were devoted to leveraging both semantic and topological graphs, and they only indirectly extracted the relationships between graphs via weights shared by features. To better capture the connections between nodes from these two types of graphs, this paper proposes a graph neural network dubbed Dual Low-Rank Graph AutoEncoder (DLR-GAE), which takes both semantic and topological homophily into consideration. Differing from prior works that share common weights between GCNs, the presented DLR-GAE conducts sustained exploration of low-rank information between two distinct graphs, and reconstructs adjacency matrices from learned latent factors and embeddings. In order to obtain valid adjacency matrices that meet certain conditions, we design some surrogates and projections to restrict the learned factor matrix. We compare the proposed model with state-of-the-art methods on several datasets, which demonstrates the superior accuracy of DLR-GAE in semi-supervised classification.
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Wagenpfeil, Stefan, Binh Vu, Paul Mc Kevitt, and Matthias Hemmje. "Fast and Effective Retrieval for Large Multimedia Collections." Big Data and Cognitive Computing 5, no. 3 (July 22, 2021): 33. http://dx.doi.org/10.3390/bdcc5030033.
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The indexing and retrieval of multimedia content is generally implemented by employing feature graphs. These graphs typically contain a significant number of nodes and edges to reflect the level of detail in feature detection. A higher level of detail increases the effectiveness of the results, but also leads to more complex graph structures. However, graph traversal-based algorithms for similarity are quite inefficient and computationally expensive, especially for large data structures. To deliver fast and effective retrieval especially for large multimedia collections and multimedia big data, an efficient similarity algorithm for large graphs in particular is desirable. Hence, in this paper, we define a graph projection into a 2D space (Graph Code) and the corresponding algorithms for indexing and retrieval. We show that calculations in this space can be performed more efficiently than graph traversals due to the simpler processing model and the high level of parallelization. As a consequence, we demonstrate experimentally that the effectiveness of retrieval also increases substantially, as the Graph Code facilitates more levels of detail in feature fusion. These levels of detail also support an increased trust prediction, particularly for fused social media content. In our mathematical model, we define a metric triple for the Graph Code, which also enhances the ranked result representations. Thus, Graph Codes provide a significant increase in efficiency and effectiveness, especially for multimedia indexing and retrieval, and can be applied to images, videos, text and social media information.
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Cao, Keyan, and Chuang Zheng. "TBRm: A Time Representation Method for Industrial Knowledge Graph." Applied Sciences 12, no. 22 (November 8, 2022): 11316. http://dx.doi.org/10.3390/app122211316.
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With the development of the artificial intelligence industry, Knowledge Graph (KG), as a concise and intuitive data presentation form, has received extensive attention and research from both academia and industry in recent years. At the same time, developments in the Internet of Things (IoT) have empowered modern industries to implement large-scale IoT ecosystems, such as the Industrial Internet of Things (IIoT). Using knowledge graphs (KG) to process data from the Industrial Internet of Things (IIoT) is a research field worthy of attention, but most of the researched knowledge graph technologies are mainly concentrated in the field of static knowledge graphs, which are composed of triples. In fact, many graphs also contain some dynamic information, such as time changes at points and time changes at edges; such knowledge graphs are called Temporal Knowledge Graphs (TKGs). We consider the temporal knowledge graph based on the projection and change of space. In order to combine the temporal information, we propose a new representation of the temporal knowledge graph, namely TBRm, which increases the temporal dimension of the translational distance model and utilizes relational predicates in time add representation in time dimension. We evaluate the proposed method on knowledge graph completion tasks using four benchmark datasets. Experiments demonstrate the effectiveness of TBRm representation in the temporal dimension. At the same time, it is also practiced on a network security data set of the Industrial Internet of Things. The practical results prove that the TBRm method can achieve good performance in terms of the degree of harm to IIoT network security.
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Zhang, Zhao Yang, Zheng Tian, and Wei Dong Yan. "Spectral Feature Matching Based on Isometric Projection of Matrix." Applied Mechanics and Materials 121-126 (October 2011): 4161–65. http://dx.doi.org/10.4028/www.scientific.net/amm.121-126.4161.
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This paper presents a spectral method to matching a pair of feature sets based on isometric projection of matrix. In the proposed method, a graph is constructed to model the structure relationships between features. Then the correspondence is found by minimizing the inner product between two isometric projections of the weighted adjacency matrix of graph. Finally, transformation between the two feature sets is estimated according to correct correspondences. The performance of the proposed approach is better than the state-of-the-art method in terms of correct ratio under position perturbation and computation time. Experiments on a number of simulated data, synthetic and real-world images show the validity of the proposed algorithm.
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CHOWDHURY, SUBHADIP. "Ziggurat fringes are self-similar." Ergodic Theory and Dynamical Systems 37, no. 3 (December 28, 2015): 739–57. http://dx.doi.org/10.1017/etds.2015.75.
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We give explicit formulae for fringe lengths of the Calegari–Walker ziggurats—i.e., graphs of extremal rotation numbers associated with positive words in free groups. These formulae reveal (partial) integral projective self-similarity in ziggurat fringes, which are low-dimensional projections of characteristic polyhedra on the bounded cohomology of free groups. This explains phenomena observed experimentally by Gordenko, Calegari and Walker.
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GARDINER, A., CHERYL E. PRAEGER, and SANMING ZHOU. "CROSS RATIO GRAPHS." Journal of the London Mathematical Society 64, no. 2 (October 2001): 257–72. http://dx.doi.org/10.1112/s0024610701002150.
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A family of arc-transitive graphs is studied. The vertices of these graphs are ordered pairs of distinct points from a finite projective line, and adjacency is defined in terms of the cross ratio. A uniform description of the graphs is given, their automorphism groups are determined, the problem of isomorphism between graphs in the family is solved, some combinatorial properties are explored, and the graphs are characterised as a certain class of arc-transitive graphs. Some of these graphs have arisen as examples in studies of arc-transitive graphs with complete quotients and arc-transitive graphs which ‘almost cover’ a 2-arc transitive graph.
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Lin, Lin, Jie Liu, Feng Guo, Changsheng Tong, Lizheng Zu, and Hao Guo. "ERDERP: Entity and Relation Double Embedding on Relation Hyperplanes and Relation Projection Hyperplanes." Mathematics 10, no. 22 (November 9, 2022): 4182. http://dx.doi.org/10.3390/math10224182.
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Since data are gradually enriched over time, knowledge graphs are inherently imperfect. Thus, knowledge graph completion is proposed to perfect knowledge graph by completing triples. Currently, a family of translation models has become the most effective method for knowledge graph completion. These translation models are modeled to solve the complexity and diversity of entities, such as one-to-many, many-to-one, and many-to-many, which ignores the diversity of relations themselves, such as multiple relations between a pair of entities. As a result, with current translation models, it is difficult to effectively extract the semantic information of entities and relations. To effectively extract the semantic information of the knowledge graph, this paper fundamentally analyzes the complex relationships of the knowledge graph. Then, considering the diversity of relations themselves, the complex relationships are refined as one-to-one-to-many, many-to-one-to-one, one-to-many-to-one, many-to-one-to-many, many-to-many-to-one, one-to-many-to-many, and many-to-many-to-many. By analyzing the complex relationships, a novel knowledge graph completion model, entity and relation double embedding on relation hyperplanes and relation projection hyperplanes (ERDERP), is proposed to extract the semantic information of entities and relations. First, ERDERP establishes a relation hyperplane for each relation and projects the relation embedding into the relation hyperplane. Thus, the semantic information of the relations is extracted effectively. Second, ERDERP establishes a relation projection hyperplane for each relation projection and projects entities into relation projection hyperplane. Thus, the semantic information of the entities is extracted effectively. Moreover, it is theoretically proved that ERDERP can solve antisymmetric problems. Finally, the proposed ERDERP are compared with several typical knowledge graph completion models. The experimental results show that ERDERP is significantly effective in link prediction, especially in relation prediction. For instance, on FB15k and FB15k-237, Hits@1 of ERDERP outperforms TransH at least 30%.
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Zinova, Polina Aleksandrovna, and Maxim Eduardovich Kazarian. "Algebra of shares, complete bipartite graphs and $\mathfrak{sl}_2$ weight system." Sbornik: Mathematics 214, no. 6 (2023): 832–52. http://dx.doi.org/10.4213/sm9795e.
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A function of chord diagrams is called a weight system if it satisfies the so-called four-term relations. Vassiliev's theory describes finite-order knot invariants in terms of weight systems. In particular, there is a weight system corresponding to the coloured Jones polynomial. This weight system is described in terms of the Lie algebra $\mathfrak{sl}_2$. According to the Chmutov-Lando theorem, the value of this weight system depends only on the intersection graph of the chord diagram. Therefore, it is possible to discuss the values of this weight system at intersection graphs. We obtain formulae for the generating functions of the values of the $\mathfrak{sl}_2$ weight system at complete bipartite graphs. Using these formulae we prove that Lando's conjecture about the degree of the polynomial that is the value of this weight system at the projection of a graph onto the subspace of primitive elements in the Hopf algebra of graphs is true for complete bipartite graphs and for a certain wider class of graphs. We introduce the algebra of shares and the $\mathfrak{sl}_2$ weight system on shares. These are the main tools for our proof. Bibliography: 14 titles.
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Wang, Peng, Jingju Liu, Dongdong Hou, and Shicheng Zhou. "A Cybersecurity Knowledge Graph Completion Method Based on Ensemble Learning and Adversarial Training." Applied Sciences 12, no. 24 (December 16, 2022): 12947. http://dx.doi.org/10.3390/app122412947.
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The application of cybersecurity knowledge graphs is attracting increasing attention. However, many cybersecurity knowledge graphs are incomplete due to the sparsity of cybersecurity knowledge. Existing knowledge graph completion methods do not perform well in domain knowledge, and they are not robust enough relative to noise data. To address these challenges, in this paper we develop a new knowledge graph completion method called CSEA based on ensemble learning and adversarial training. Specifically, we integrate a variety of projection and rotation operations to model the relationships between entities, and use angular information to distinguish entities. A cooperative adversarial training method is designed to enhance the generalization and robustness of the model. We combine the method of generating perturbations for the embedding layers with the self-adversarial training method. The UCB (upper confidence bound) multi-armed bandit method is used to select the perturbations of the embedding layer. This achieves a balance between perturbation diversity and maximum loss. To this end, we build a cybersecurity knowledge graph based on the CVE, CWE, and CAPEC cybersecurity databases. Our experimental results demonstrate the superiority of our proposed model for completing cybersecurity knowledge graphs.