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Arifin, Samsul, Indra Bayu Muktyas, and Jeremy Matthew Mandei. "Graph coloring program for variation of exam scheduling modeling at Binus University based on Welsh and Powell algorithm." Journal of Physics: Conference Series 2279, no. 1 (May 1, 2022): 012005. http://dx.doi.org/10.1088/1742-6596/2279/1/012005.
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Abstract In this article, we will design a computer program to solve graph colouring and to visualize the variation of exam scheduling modelling at Binus University in graphs based on the Welch and Powell Algorithm. A vertex colouring of graph G = (V, E) is an F: V → N mapping where the adjacent vertex are different colours in N, i.e. if the edge (u, v) are in E, then F (u) is not equal to F (v). The formulation of problems regarding computer programs based on a small experiment and variations in modelling exams is carried out in graph form using the Visual Basic Application.
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Firmansah, Fery. "Pelabelan Harmonis Ganjil pada Graf Bunga Double Quadrilateral." JURNAL ILMIAH SAINS 20, no. 1 (March 10, 2020): 12. http://dx.doi.org/10.35799/jis.20.1.2020.27278.
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Graf harmonis ganjil adalah graf yang memenuhi sifat-sifat pelabelan harmonis ganjil. Tujuan dari penelitian ini adalah mendapatkan kelas graf baru yang merupakan graf harmonis ganjil. Metode penelitian yang digunakan terdiri dari beberapa tahapan yaitu konstruksi definisi, formulasi fungsi pelabelan dan pembuktian teorema. Hasil dari penelitian ini adalah konstruksi graf bunga double quadrilateral dengan dan graf bunga variasi double quadrilateral dengan yang merupakan pengembangan dari graf double quadrilateral dan graf variasi double quadrilateral . Lebih lanjut telah dibuktikan bahwa graf dan adalah graf harmonis ganjil.Kata Kunci: graf double quadrilateral, graf bunga, graf harmonis ganjil, pelabelan graf Odd Harmonious Labelling on The Flower Double Quadrilateral Graphs ABSTRACTOdd harmonious graphs are graphs that have odd harmonious labeling properties. The purpose of this study is to get a new class of graphs which are odd harmonious graphs. The research method used consists of several stages, namely construction of definitions, formulation of labeling functions and proof of theorems. The results of this study is to get a graph construction will be given, namely the flower quadrilateral graphs with and the flower variation of quadrilateral graphs with , which are the development of double quadrilateral graphs and variation double quadrilateral graphs . It has further been proven that and are odd harmonious graphs.Keywords: double quadrilateral graph, flower graph, labeling graph, odd harmonious graph
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Prabhakaran, Rahul, Giovanni Bertotti, Janos Urai, and David Smeulders. "Investigating spatial heterogeneity within fracture networks using hierarchical clustering and graph distance metrics." Solid Earth 12, no. 10 (September 30, 2021): 2159–209. http://dx.doi.org/10.5194/se-12-2159-2021.
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Abstract. Rock fractures organize as networks, exhibiting natural variation in their spatial arrangements. Therefore, identifying, quantifying, and comparing variations in spatial arrangements within network geometries are of interest when explicit fracture representations or discrete fracture network models are chosen to capture the influence of fractures on bulk rock behaviour. Treating fracture networks as spatial graphs, we introduce a novel approach to quantify spatial variation. The method combines graph similarity measures with hierarchical clustering and is applied to investigate the spatial variation within large-scale 2-D fracture networks digitized from the well-known Lilstock limestone pavements, Bristol Channel, UK. We consider three large, fractured regions, comprising nearly 300 000 fractures spread over 14 200 m2 from the Lilstock pavements. Using a moving-window sampling approach, we first subsample the large networks into subgraphs. Four graph similarity measures – fingerprint distance, D-measure, Network Laplacian spectral descriptor (NetLSD), and portrait divergence – that encapsulate topological relationships and geometry of fracture networks are then used to compute pair-wise subgraph distances serving as input for the statistical hierarchical clustering technique. In the form of hierarchical dendrograms and derived spatial variation maps, the results indicate spatial autocorrelation with localized spatial clusters that gradually vary over distances of tens of metres with visually discernable and quantifiable boundaries. Fractures within the identified clusters exhibit differences in fracture orientations and topology. The comparison of graph similarity-derived clusters with fracture persistence measures indicates an intra-network spatial variation that is not immediately obvious from the ubiquitous fracture intensity and density maps. The proposed method provides a quantitative way to identify spatial variations in fracture networks, guiding stochastic and geostatistical approaches to fracture network modelling.
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Rautiainen, Mikko, Veli Mäkinen, and Tobias Marschall. "Bit-parallel sequence-to-graph alignment." Bioinformatics 35, no. 19 (March 9, 2019): 3599–607. http://dx.doi.org/10.1093/bioinformatics/btz162.
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Abstract Motivation Graphs are commonly used to represent sets of sequences. Either edges or nodes can be labeled by sequences, so that each path in the graph spells a concatenated sequence. Examples include graphs to represent genome assemblies, such as string graphs and de Bruijn graphs, and graphs to represent a pan-genome and hence the genetic variation present in a population. Being able to align sequencing reads to such graphs is a key step for many analyses and its applications include genome assembly, read error correction and variant calling with respect to a variation graph. Results We generalize two linear sequence-to-sequence algorithms to graphs: the Shift-And algorithm for exact matching and Myers’ bitvector algorithm for semi-global alignment. These linear algorithms are both based on processing w sequence characters with a constant number of operations, where w is the word size of the machine (commonly 64), and achieve a speedup of up to w over naive algorithms. For a graph with |V| nodes and |E| edges and a sequence of length m, our bitvector-based graph alignment algorithm reaches a worst case runtime of O(|V|+⌈mw⌉|E| log w) for acyclic graphs and O(|V|+m|E| log w) for arbitrary cyclic graphs. We apply it to five different types of graphs and observe a speedup between 3-fold and 20-fold compared with a previous (asymptotically optimal) alignment algorithm. Availability and implementation https://github.com/maickrau/GraphAligner Supplementary information Supplementary data are available at Bioinformatics online.
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Bhat, Farooq Ahmad, and M. Arif Wani. "Elastic Bunch Graph Matching Based Face Recognition Under Varying Lighting, Pose, and Expression Conditions." IAES International Journal of Artificial Intelligence (IJ-AI) 3, no. 4 (August 20, 2016): 177. http://dx.doi.org/10.11591/ijai.v3.i4.pp177-182.
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In this paper performance of elastic bunch graph matching (EBGM) for face recognition under variation in facial expression, variation in lighting condition and variation in poses are given. In this approach faces are represented by labelled graphs. Experimental results of EBGM on ORL, Yale B and FERET datasets are provided. Strong and weak features of EBGM algorithm are discussed.
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Qiu, Yutong, and Carl Kingsford. "Constructing small genome graphs via string compression." Bioinformatics 37, Supplement_1 (July 1, 2021): i205—i213. http://dx.doi.org/10.1093/bioinformatics/btab281.
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Abstract Motivation The size of a genome graph—the space required to store the nodes, node labels and edges—affects the efficiency of operations performed on it. For example, the time complexity to align a sequence to a graph without a graph index depends on the total number of characters in the node labels and the number of edges in the graph. This raises the need for approaches to construct space-efficient genome graphs. Results We point out similarities in the string encoding mechanisms of genome graphs and the external pointer macro (EPM) compression model. We present a pair of linear-time algorithms that transform between genome graphs and EPM-compressed forms. The algorithms result in an upper bound on the size of the genome graph constructed in terms of an optimal EPM compression. To further reduce the size of the genome graph, we propose the source assignment problem that optimizes over the equivalent choices during compression and introduce an ILP formulation that solves that problem optimally. As a proof-of-concept, we introduce RLZ-Graph, a genome graph constructed based on the relative Lempel–Ziv algorithm. Using RLZ-Graph, across all human chromosomes, we are able to reduce the disk space to store a genome graph on average by 40.7% compared to colored compacted de Bruijn graphs constructed by Bifrost under the default settings. The RLZ-Graph scales well in terms of running time and graph sizes with an increasing number of human genome sequences compared to Bifrost and variation graphs produced by VGtoolkit. Availability The RLZ-Graph software is available at: https://github.com/Kingsford-Group/rlzgraph. Supplementary information Supplementary data are available at Bioinformatics online.
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Angadi, Shanmukhappa A., and Sanjeevakumar M. Hatture. "Face Recognition Through Symbolic Modeling of Face Graphs and Texture." International Journal of Pattern Recognition and Artificial Intelligence 33, no. 12 (November 2019): 1956008. http://dx.doi.org/10.1142/s0218001419560081.
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Face recognition helps in authentication of the user using remotely acquired facial information. The dynamic nature of face images like pose, illumination, expression, occlusion, aging, etc. degrades the performance of the face recognition system. In this paper, a new face recognition system using facial images with illumination variation, pose variation and partial occlusion is presented. The facial image is described as a collection of three complete connected graphs and these graphs are represented as symbolic objects. The structural characteristics, i.e. graph spectral properties, energy of graph, are extracted and embedded in a symbolic object. The texture features from the cheeks portions are extracted using center symmetric local binary pattern (CS-LBP) descriptor. The global features of the face image, i.e. length and width, are also extracted. Further symbolic data structure is constructed using the above features, namely, the graph spectral properties, energy of graph, global features and texture features. User authentication is performed using a new symbolic similarity metric. The performance is investigated by conducting the experiments with AR face database and VTU-BEC-DB multimodal database. The experimental results demonstrate an identification rate of 95.97% and 97.20% for the two databases.
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Youssef, Maged Zakaria, and Zainab Saad Almoreed. "On odd prime labeling of graphs." Open Journal of Discrete Applied Mathematics 3, no. 3 (October 20, 2020): 33–40. http://dx.doi.org/10.30538/psrp-odam2020.0041.
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In this paper we give a new variation of the prime labeling. We call a graph \(G\) with vertex set \(V(G)\) has an odd prime labeling if its vertices can be labeled distinctly from the set \(\big\{1, 3, 5, ...,2\big|V(G)\big| -1\big\}\) such that for every edge \(xy\) of \(E(G)\) the labels assigned to the vertices of \(x\) and \(y\) are relatively prime. A graph that admits an odd prime labeling is called an <i>odd prime graph</i>. We give some families of odd prime graphs and give some necessary conditions for a graph to be odd prime. Finally, we conjecture that every prime graph is odd prime graph.
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Kuhlmann, Marco, and Stephan Oepen. "Towards a Catalogue of Linguistic Graph Banks." Computational Linguistics 42, no. 4 (December 2016): 819–27. http://dx.doi.org/10.1162/coli_a_00268.
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Graphs exceeding the formal complexity of rooted trees are of growing relevance to much NLP research. Although formally well understood in graph theory, there is substantial variation in the types of linguistic graphs, as well as in the interpretation of various structural properties. To provide a common terminology and transparent statistics across different collections of graphs in NLP, we propose to establish a shared community resource with an open-source reference implementation for common statistics.
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Da Silva, Kévin, Nicolas Pons, Magali Berland, Florian Plaza Oñate, Mathieu Almeida, and Pierre Peterlongo. "StrainFLAIR: strain-level profiling of metagenomic samples using variation graphs." PeerJ 9 (August 23, 2021): e11884. http://dx.doi.org/10.7717/peerj.11884.
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Current studies are shifting from the use of single linear references to representation of multiple genomes organised in pangenome graphs or variation graphs. Meanwhile, in metagenomic samples, resolving strain-level abundances is a major step in microbiome studies, as associations between strain variants and phenotype are of great interest for diagnostic and therapeutic purposes. We developed StrainFLAIR with the aim of showing the feasibility of using variation graphs for indexing highly similar genomic sequences up to the strain level, and for characterizing a set of unknown sequenced genomes by querying this graph. On simulated data composed of mixtures of strains from the same bacterial species Escherichia coli, results show that StrainFLAIR was able to distinguish and estimate the abundances of close strains, as well as to highlight the presence of a new strain close to a referenced one and to estimate its abundance. On a real dataset composed of a mix of several bacterial species and several strains for the same species, results show that in a more complex configuration StrainFLAIR correctly estimates the abundance of each strain. Hence, results demonstrated how graph representation of multiple close genomes can be used as a reference to characterize a sample at the strain level.
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Bigham, Abigail, Elizabeth Ann Donovan, James Pack, Jordan Turley, and Lesley Wiglesworth. "Prime Labelings of Snake Graphs." PUMP Journal of Undergraduate Research 2 (August 21, 2019): 131–49. http://dx.doi.org/10.46787/pump.v2i0.1274.
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A prime labeling of a graph G with n vertices is a labeling of the vertices with distinct integers from the set {1, 2 ,..., n} such that the labels of any two adjacent vertices are relatively prime. In this paper, we introduce a snake graph, the fused union of identical cycles, and define a consecutive snake prime labeling for this new family of graphs. We characterize some snake graphs that have a consecutive snake prime labeling and then consider a variation of this labeling.
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Jain, Chirag, Neda Tavakoli, and Srinivas Aluru. "A variant selection framework for genome graphs." Bioinformatics 37, Supplement_1 (July 1, 2021): i460—i467. http://dx.doi.org/10.1093/bioinformatics/btab302.
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Abstract Motivation Variation graph representations are projected to either replace or supplement conventional single genome references due to their ability to capture population genetic diversity and reduce reference bias. Vast catalogues of genetic variants for many species now exist, and it is natural to ask which among these are crucial to circumvent reference bias during read mapping. Results In this work, we propose a novel mathematical framework for variant selection, by casting it in terms of minimizing variation graph size subject to preserving paths of length α with at most δ differences. This framework leads to a rich set of problems based on the types of variants [e.g. single nucleotide polymorphisms (SNPs), indels or structural variants (SVs)], and whether the goal is to minimize the number of positions at which variants are listed or to minimize the total number of variants listed. We classify the computational complexity of these problems and provide efficient algorithms along with their software implementation when feasible. We empirically evaluate the magnitude of graph reduction achieved in human chromosome variation graphs using multiple α and δ parameter values corresponding to short and long-read resequencing characteristics. When our algorithm is run with parameter settings amenable to long-read mapping (α = 10 kbp, δ = 1000), 99.99% SNPs and 73% SVs can be safely excluded from human chromosome 1 variation graph. The graph size reduction can benefit downstream pan-genome analysis. Availability and implementation https://github.com/AT-CG/VF. Supplementary information Supplementary data are available at Bioinformatics online.
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Benediktovich, V. I. "Analogue of Brauer’s conjecture for the signless Laplacian of cographs." Proceedings of the National Academy of Sciences of Belarus. Physics and Mathematics Series 56, no. 3 (October 18, 2020): 310–17. http://dx.doi.org/10.29235/1561-2430-2020-56-3-310-317.
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In this paper, we consider the class of cographs and its subclasses, namely, threshold graphs and anti-regular graphs. In 2011 H. Bai confirmed the Grone – Merris conjecture about the sum of the first k eigenvalues of the Laplacian of an arbitrary graph. As a variation of the Grone – Merris conjecture, A. Brouwer put forward his conjecture about an upper bound for this sum. Although the latter conjecture was confirmed for many graph classes, however, it remains open. By analogy to Brouwer’s conjecture, in 2013 F. Ashraf et al. put forward a conjecture about the sum of k eigenvalues of the signless Laplacian, which was also confirmed for some graph classes but remains open. In this paper, an analogue of the Brouwer’s conjecture is confirmed for the graph classes under our consideration for the eigenvalues of their signless Laplacian for some natural k which does not exceed the order of the considered graphs.
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Liu, Feng, and Xiao Zhang. "Variation Inequalities for the Hardy-Littlewood Maximal Function on Finite Directed Graphs." Mathematics 10, no. 6 (March 16, 2022): 950. http://dx.doi.org/10.3390/math10060950.
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In this paper, the authors establish the bounds for the Hardy-Littlewood maximal operator defined on a finite directed graph G→ in the space BVp(G→) of bounded p-variation functions. More precisely, the authors obtain the BVp norms of MG→ for some directed graphs G→.
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Karuza, Elisabeth A., Ari E. Kahn, and Danielle S. Bassett. "Human Sensitivity to Community Structure Is Robust to Topological Variation." Complexity 2019 (February 11, 2019): 1–8. http://dx.doi.org/10.1155/2019/8379321.
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Despite mounting evidence that human learners are sensitive to community structure underpinning temporal sequences, this phenomenon has been studied using an extremely narrow set of network ensembles. The extent to which behavioral signatures of learning are robust to changes in community size and number is the focus of the present work. Here we present adult participants with a continuous stream of novel objects generated by a random walk along graphs of 1, 2, 3, 4, or 6 communities comprised of N = 24, 12, 8, 6, and 4 nodes, respectively. Nodes of the graph correspond to a unique object and edges correspond to their immediate succession in the stream. In short, we find that previously observed processing costs associated with community boundaries persist across an array of graph architectures. These results indicate that statistical learning mechanisms can flexibly accommodate variation in community structure during visual event segmentation.
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Meinawati, Rima, Vika Yugi Kurniawan, and Nughthoh Arfawi Kurdhi. "Algorithm for Constructing Total Graph of Commutative Ring." JTAM (Jurnal Teori dan Aplikasi Matematika) 8, no. 2 (March 30, 2024): 351. http://dx.doi.org/10.31764/jtam.v8i2.19850.
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Let R be a commutative ring. The total graph of R, denoted by TΓ(R) is a graph whose vertices are all elements of the ring R and every i,j∈R with i≠j, then i and j vertices are connected by edges if and only if i+j∈Z(R), where Z(R) is the set of zero-divisors in R with 0∈Z(R). Python programming is code that is easy to learn, read, understand, and helpful in explaining problems regarding graphs and algebra. In this paper, we determine an algorithm to construct the total graph of ring Z_n using Python. The research methods in this paper is a literature studies. The results generated by the algorithm can be utilized to observe the characteristic patterns displayed by the graph. Based on the algorithm’s constructed graph pattern, several properties of TΓ(Z_n ) can be inferred. For instance, if n is a prime number, then TΓ(Z_n ) is a disconnected graph. On the other hand, if n is a prime number and n≥3, then TΓ(Z_2n ) and TΓ(Z_4n ) is a connected graph, regular graph, Hamiltonian graph, and has a girth gr(TΓ(〖Z〗_n ))=3. In this paper we creating an algorithm to construct total graphs from commutative rings streamlines the construction process, enhances accessibility and utilization of total graphs, and supports parameter variation exploration and application in problem-solving.
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Chen, Hanlin, Renfang Wu, Guihua Huang, and Hanyuan Deng. "Dimer–monomer model on the Towers of Hanoi graphs." International Journal of Modern Physics B 29, no. 23 (September 17, 2015): 1550173. http://dx.doi.org/10.1142/s0217979215501738.
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The number of dimer–monomers (matchings) of a graph [Formula: see text] is an important graph parameter in statistical physics. Following recent research, we study the asymptotic behavior of the number of dimer–monomers [Formula: see text] on the Towers of Hanoi graphs and another variation of the Sierpiński graphs which is similar to the Towers of Hanoi graphs, and derive the recursion relations for the numbers of dimer–monomers. Upper and lower bounds for the entropy per site, defined as [Formula: see text], where [Formula: see text] is the number of vertices in a graph [Formula: see text], on these Sierpiński graphs are derived in terms of the numbers at a certain stage. As the difference between these bounds converges quickly to zero as the calculated stage increases, the numerical value of the entropy can be evaluated with more than a hundred significant figures accuracy.
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Christensen, Andrew J., Ananya Sen Gupta, and Ivars Kirsteins. "Graph representation learning on braid manifolds." Journal of the Acoustical Society of America 152, no. 4 (October 2022): A39. http://dx.doi.org/10.1121/10.0015466.
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The accuracy of autonomous sonar target recognition systems is usually hindered by morphing target features, unknown target geometry, and uncertainty caused by waveguide distortions to signal. Common “black-box” neural networks are not effective in addressing these challenges since they do not produce physically interpretable features. This work seeks to use recent advancements in machine learning to extract braid features that can be interpreted by a domain expert. We utilize Graph Neural Networks (GNNs) to discover braid manifolds in sonar ping spectra data. This approach represents the sonar ping data as a sequence of timestamped, sparse, dynamic graphs. These dynamic graph sequences are used as input into a GNN to produce feature dictionaries. GNNs ability to learn on complex systems of interactions help make them resilient to environmental uncertainty. To learn the evolving braid-like features of the sonar ping spectra graphs, a modified variation of Temporal Graph Networks (TGNs) is used. TGNs can perform prediction and classification tasks on timestamped dynamic graphs. The modified TGN in this work models the evolution of the sonar ping spectra graph to eventually perform graph-based classification. [Work supported by ONR grant N00014-21-1-2420.]
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Tang, Xiangyun, Dongliang Liao, Weijie Huang, Jin Xu, Liehuang Zhu, and Meng Shen. "Fully Exploiting Cascade Graphs for Real-time Forwarding Prediction." Proceedings of the AAAI Conference on Artificial Intelligence 35, no. 1 (May 18, 2021): 582–90. http://dx.doi.org/10.1609/aaai.v35i1.16137.
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Real-time forwarding prediction for predicting online contents' popularity is beneficial to various social applications for enhancing interactive social behaviors. Cascade graphs, formed by online contents' propagation, play a vital role in real-time forwarding prediction. Existing cascade graph modeling methods are inadequate to embed cascade graphs that have hub structures and deep cascade paths, or they fail to handle the short-term outbreak of forwarding amount. To this end, we propose a novel real-time forwarding prediction method that includes an effective approach for cascade graph embedding and a short-term variation sensitive method for time-series modeling, making the best of cascade graph features. Using two real world datasets, we demonstrate the significant superiority of the proposed method compared with the state-of-the-art. Our experiments also reveal interesting implications hidden in the performance differences between cascade graph embedding and time-series modeling.
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Caselli, A., G. Falquet, and C. Métral. "KNOWLEDGE GRAPH CONSTRUCTION FOR SUBSURFACE OBJECTS INCLUDING UNCERTAINTY AND TIME VARIATION." International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences XLVI-4/W4-2021 (October 7, 2021): 131–36. http://dx.doi.org/10.5194/isprs-archives-xlvi-4-w4-2021-131-2021.
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Abstract. In the recent years the concept of knowledge graph has emerged as a way to aggregate information from various sources without imposing too strict data modelling constraints. Several graph models have been proposed during the years, ranging from the “standard” RDF to more expressive ones, such as Neo4J and RDF-star. The adoption of knowledge graph has become established in several domains. It is for instance the case of the 3D geoinformation domain, where the adoption of semantic web technologies has led to several works in data integration and publishing. However, yet there is not a well-defined model or technique to represent 3D geoinformation including uncertainty and time variation in knowledge graphs. In this paper we propose a model to represent parameterized geometries of subsurface objects. The vocabulary of the model has been defined as an OWL ontology and it extends existing ontologies by adding classes and properties to represent the uncertainty and the spatio-temporal behaviour of a geometry, as well as additional attributes, such as the data provenance. The model has been validated on significant use cases showing different types of uncertainties on 3D subsurface objects. A possible implementation is also presented, using RDF-star for the data representation.
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Mo, Yujie, Liang Peng, Jie Xu, Xiaoshuang Shi, and Xiaofeng Zhu. "Simple Unsupervised Graph Representation Learning." Proceedings of the AAAI Conference on Artificial Intelligence 36, no. 7 (June 28, 2022): 7797–805. http://dx.doi.org/10.1609/aaai.v36i7.20748.
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In this paper, we propose a simple unsupervised graph representation learning method to conduct effective and efficient contrastive learning. Specifically, the proposed multiplet loss explores the complementary information between the structural information and neighbor information to enlarge the inter-class variation, as well as adds an upper bound loss to achieve the finite distance between positive embeddings and anchor embeddings for reducing the intra-class variation. As a result, both enlarging inter-class variation and reducing intra-class variation result in small generalization error, thereby obtaining an effective model. Furthermore, our method removes widely used data augmentation and discriminator from previous graph contrastive learning methods, meanwhile available to output low-dimensional embeddings, leading to an efficient model. Experimental results on various real-world datasets demonstrate the effectiveness and efficiency of our method, compared to state-of-the-art methods. The source codes are released at https://github.com/YujieMo/SUGRL.
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Ko, Eun-Sung. "A study on teaching of reasoning about variation: Focusing on statistics unit in 3rd and 4th grade of elementary school." Korean Association For Learner-Centered Curriculum And Instruction 23, no. 12 (June 30, 2023): 733–49. http://dx.doi.org/10.22251/jlcci.2023.23.12.733.
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Objectives Although reasoning about variation is important in statistical thinking, little research has been conducted on how reasoning about variation is taught in the statistics unit of school mathematics. In this study, we will look into the teaching of reasoning about variation in the statistics unit of the 3rd and 4th grade mathematics textbooks and discuss methods to strengthen the teaching of reasoning about variation.
Methods In this study, the activities provided for reasoning about variation in the statistics unit were analyzed. To this end, the activities provided in the statistics unit of the 3rd and 4th grade elementary school mathematics textbooks (state 1 type, accredited 10 types) are divided into 4 stages of statistical problem solving (problem formulation, data collection, data analysis, data interpretation), and the teaching of reasoning about variation in each stage is analyzed.
Results In the problem formulation stage, all three units(picture graph, bar graph, and line graph) provided activities to deal with the variability inherent in the problem situation. In the case of the data collection stage, there was a difference for each unit. Activities to consider the effect of data collection methods on variability and activities to consider how selection items during investigation affect data variability were provided. In the data analysis stage, activities to recognize the merits of graphs, activities to identify trends in data, and activities to compare graphs with different units were provided. In the data interpretation stage, activities to compare the two groups, activities to make decisions based on data, and activities to explain changes in data were provided.
Conclusions In this study, it was confirmed that the teaching of reasoning about variation in the data collection stage was considerably poor compared to other stages. Continuous research should be conducted on the development of problem situations, activities, and questions suitable for the level of students for the teaching of reasoning about variation.
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Chang, Xian, Jordan Eizenga, Adam M. Novak, Jouni Sirén, and Benedict Paten. "Distance indexing and seed clustering in sequence graphs." Bioinformatics 36, Supplement_1 (July 1, 2020): i146—i153. http://dx.doi.org/10.1093/bioinformatics/btaa446.
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Abstract Motivation Graph representations of genomes are capable of expressing more genetic variation and can therefore better represent a population than standard linear genomes. However, due to the greater complexity of genome graphs relative to linear genomes, some functions that are trivial on linear genomes become much more difficult in genome graphs. Calculating distance is one such function that is simple in a linear genome but complicated in a graph context. In read mapping algorithms such distance calculations are fundamental to determining if seed alignments could belong to the same mapping. Results We have developed an algorithm for quickly calculating the minimum distance between positions on a sequence graph using a minimum distance index. We have also developed an algorithm that uses the distance index to cluster seeds on a graph. We demonstrate that our implementations of these algorithms are efficient and practical to use for a new generation of mapping algorithms based upon genome graphs. Availability and implementation Our algorithms have been implemented as part of the vg toolkit and are available at https://github.com/vgteam/vg.
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Oggier, Frédérique, and Anwitaman Datta. "Renyi entropy driven hierarchical graph clustering." PeerJ Computer Science 7 (February 25, 2021): e366. http://dx.doi.org/10.7717/peerj-cs.366.
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This article explores a graph clustering method that is derived from an information theoretic method that clusters points in ${{\mathbb{R}}^{n}}$ relying on Renyi entropy, which involves computing the usual Euclidean distance between these points. Two view points are adopted: (1) the graph to be clustered is first embedded into ${\mathbb{R}}^{d}$ for some dimension d so as to minimize the distortion of the embedding, then the resulting points are clustered, and (2) the graph is clustered directly, using as distance the shortest path distance for undirected graphs, and a variation of the Jaccard distance for directed graphs. In both cases, a hierarchical approach is adopted, where both the initial clustering and the agglomeration steps are computed using Renyi entropy derived evaluation functions. Numerical examples are provided to support the study, showing the consistency of both approaches (evaluated in terms of F-scores).
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Ko, Sung Moon, Sungjun Cho, Dae-Woong Jeong, Sehui Han, Moontae Lee, and Honglak Lee. "Grouping Matrix Based Graph Pooling with Adaptive Number of Clusters." Proceedings of the AAAI Conference on Artificial Intelligence 37, no. 7 (June 26, 2023): 8334–42. http://dx.doi.org/10.1609/aaai.v37i7.26005.
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Graph pooling is a crucial operation for encoding hierarchical structures within graphs. Most existing graph pooling approaches formulate the problem as a node clustering task which effectively captures the graph topology. Conventional methods ask users to specify an appropriate number of clusters as a hyperparameter, then assuming that all input graphs share the same number of clusters. In inductive settings where the number of clusters could vary, however, the model should be able to represent this variation in its pooling layers in order to learn suitable clusters. Thus we propose GMPool, a novel differentiable graph pooling architecture that automatically determines the appropriate number of clusters based on the input data. The main intuition involves a grouping matrix defined as a quadratic form of the pooling operator, which induces use of binary classification probabilities of pairwise combinations of nodes. GMPool obtains the pooling operator by first computing the grouping matrix, then decomposing it. Extensive evaluations on molecular property prediction tasks demonstrate that our method outperforms conventional methods.
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Martynov, N. N., O. V. Khandarova, and F. V. Khandarov. "Graph Clustering Based on Modularity Variation Estimations." Bulletin of Irkutsk State University. Series Mathematics 25 (2018): 63–78. http://dx.doi.org/10.26516/1997-7670.2018.25.63.
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Shetty, Archana, Shubha H. V. Shubha H V, Vijaya Chowdappa, and Vivek T. G. Vivek T G. "A Study of Variation in Adult Thrombocytopenic Histograms A Graph Often Overlooked." Annals of Pathology and Laboratory Medicine 7, no. 2 (March 7, 2020): A83–88. http://dx.doi.org/10.21276/apalm.2665.
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Liu, Jianxi. "On the variation of the Randic index with given girth and leaves." Filomat 28, no. 9 (2014): 1849–53. http://dx.doi.org/10.2298/fil1409849l.
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The variation of Randic index R'(G) of a graph G is defined by R'(G) = ?uv 1/ max{du,dv}, where du is the degree of a vertex u in G and the summation extends over all edges uv of G. In this work, we characterize the extremal trees achieving the minimum value of R0 for trees with given number of vertices and leaves. Furthermore, we characterize the extremal graphs achieving the minimum value of R' for connected graphs with given number of vertices and girth.
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Qu, Ruyuan, Hui Feng, Chongbin Xu, and Bo Hu. "Analysis of Hypergraph Signals via High-Order Total Variation." Symmetry 14, no. 3 (March 7, 2022): 543. http://dx.doi.org/10.3390/sym14030543.
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Beyond pairwise relationships, interactions among groups of agents do exist in many real-world applications, but they are difficult to capture by conventional graph models. Generalized from graphs, hypergraphs have been introduced to describe such high-order group interactions. Inspired by graph signal processing (GSP) theory, an existing hypergraph signal processing (HGSP) method presented a spectral analysis framework relying on the orthogonal CP decomposition of adjacency tensors. However, such decomposition may not exist even for supersymmetric tensors. In this paper, we propose a high-order total variation (HOTV) form of a hypergraph signal (HGS) as its smoothness measure, which is a hyperedge-wise measure aggregating all signal values in each hyperedge instead of a pairwise one in most existing work. Further, we propose an HGS analysis framework based on the Tucker decomposition of the hypergraph Laplacian induced by the aforementioned HOTV. We construct an orthonormal basis from the HOTV, by which a new spectral transformation of the HGS is introduced. Then, we design hypergraph filters in both vertex and spectral domains correspondingly. Finally, we illustrate the advantages of the proposed framework by applications in label learning.
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Firmansah, Fery. "The Odd Harmonious Labeling on Variation of the Double Quadrilateral Windmill Graphs." Jurnal ILMU DASAR 18, no. 2 (October 26, 2017): 109. http://dx.doi.org/10.19184/jid.v18i2.5648.
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A Graphis calledif it hasvertices and edqes. The graphis said to be an odd harmonious if there exists an injectionsuch that induced mappingis a bijection and is said to be an odd harmonious labeling of. In this paper, we construct the odd harmonious labeling on variation of the double quadrilateral windmill graph with . Moreover, we also give odd harmonious labeling construction for generalized variation of the double quadrilateral windmill graph withand . Keywords: Odd harmonious labeling, variation, double quadrilteral, windmill graph.
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Biederstedt, Evan, Jeffrey C. Oliver, Nancy F. Hansen, Aarti Jajoo, Nathan Dunn, Andrew Olson, Ben Busby, and Alexander T. Dilthey. "NovoGraph: Genome graph construction from multiple long-read de novo assemblies." F1000Research 7 (September 3, 2018): 1391. http://dx.doi.org/10.12688/f1000research.15895.1.
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Genome graphs are emerging as an important novel approach to the analysis of high-throughput sequencing data. By explicitly representing genetic variants and alternative haplotypes in a mappable data structure, they can enable the improved analysis of structurally variable and hyperpolymorphic regions of the genome. In most existing approaches, graphs are constructed from variant call sets derived from short-read sequencing. As long-read sequencing becomes more cost-effective and enables de novo assembly for increasing numbers of whole genomes, a method for the direct construction of a genome graph from sets of assembled human genomes would be desirable. Such assembly-based genome graphs would encompass the wide spectrum of genetic variation accessible to long-read-based de novo assembly, including large structural variants and divergent haplotypes. Here we present NovoGraph, a method for the construction of a genome graph directly from a set of de novo assemblies. NovoGraph constructs a genome-wide multiple sequence alignment of all input contigs and uses a simple criterion of homologous-identical recombination to convert the multiple sequence alignment into a graph. NovoGraph outputs resulting graphs in VCF format that can be loaded into third-party genome graph toolkits. To demonstrate NovoGraph, we construct a genome graph with 23,478,835 variant sites and 30,582,795 variant alleles from de novo assemblies of seven ethnically diverse human genomes (AK1, CHM1, CHM13, HG003, HG004, HX1, NA19240). Initial evaluations show that mapping against the constructed graph reduces the average mismatch rate of reads from sample NA12878 by approximately 0.2%, albeit at a slightly increased rate of reads that remain unmapped.
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Biederstedt, Evan, Jeffrey C. Oliver, Nancy F. Hansen, Aarti Jajoo, Nathan Dunn, Andrew Olson, Ben Busby, and Alexander T. Dilthey. "NovoGraph: Human genome graph construction from multiple long-read de novo assemblies." F1000Research 7 (December 10, 2018): 1391. http://dx.doi.org/10.12688/f1000research.15895.2.
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Genome graphs are emerging as an important novel approach to the analysis of high-throughput human sequencing data. By explicitly representing genetic variants and alternative haplotypes in a mappable data structure, they can enable the improved analysis of structurally variable and hyperpolymorphic regions of the genome. In most existing approaches, graphs are constructed from variant call sets derived from short-read sequencing. As long-read sequencing becomes more cost-effective and enables de novo assembly for increasing numbers of whole genomes, a method for the direct construction of a genome graph from sets of assembled human genomes would be desirable. Such assembly-based genome graphs would encompass the wide spectrum of genetic variation accessible to long-read-based de novo assembly, including large structural variants and divergent haplotypes. Here we present NovoGraph, a method for the construction of a human genome graph directly from a set of de novo assemblies. NovoGraph constructs a genome-wide multiple sequence alignment of all input contigs and creates a graph by merging the input sequences at positions that are both homologous and sequence-identical. NovoGraph outputs resulting graphs in VCF format that can be loaded into third-party genome graph toolkits. To demonstrate NovoGraph, we construct a genome graph with 23,478,835 variant sites and 30,582,795 variant alleles from de novo assemblies of seven ethnically diverse human genomes (AK1, CHM1, CHM13, HG003, HG004, HX1, NA19240). Initial evaluations show that mapping against the constructed graph reduces the average mismatch rate of reads from sample NA12878 by approximately 0.2%, albeit at a slightly increased rate of reads that remain unmapped.
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Ackerman, Nate, Cameron E. Freer, Younesse Kaddar, Jacek Karwowski, Sean Moss, Daniel Roy, Sam Staton, and Hongseok Yang. "Probabilistic Programming Interfaces for Random Graphs: Markov Categories, Graphons, and Nominal Sets." Proceedings of the ACM on Programming Languages 8, POPL (January 5, 2024): 1819–49. http://dx.doi.org/10.1145/3632903.
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We study semantic models of probabilistic programming languages over graphs, and establish a connection to graphons from graph theory and combinatorics. We show that every well-behaved equational theory for our graph probabilistic programming language corresponds to a graphon, and conversely, every graphon arises in this way. We provide three constructions for showing that every graphon arises from an equational theory. The first is an abstract construction, using Markov categories and monoidal indeterminates. The second and third are more concrete. The second is in terms of traditional measure theoretic probability, which covers 'black-and-white' graphons. The third is in terms of probability monads on the nominal sets of Gabbay and Pitts. Specifically, we use a variation of nominal sets induced by the theory of graphs, which covers Erdős-Rényi graphons. In this way, we build new models of graph probabilistic programming from graphons.
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Christensen, Andrew J., Ananya Sen Gupta, and Ivars Kirsteins. "Sonar target feature representation using temporal graph networks." Journal of the Acoustical Society of America 151, no. 4 (April 2022): A102. http://dx.doi.org/10.1121/10.0010791.
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Autonomous sonar target recognition suffers from uncertainty caused by waveguide distortions to signal, unknown target geometry, and morphing target features. Typical “black-box” neural networks do not produce physically interpretable features and, therefore, are not effective in meeting these challenges. The primary objective of our work is to harness signal processing with machine learning to extract braided features that allow such physical interpretation by a domain expert. In this work, we introduce a feature extraction method using graph neural networks (GNNs) that seeks to discover braid manifolds from sonar magnitude spectra data. The approach involves representing the sonar magnitude spectra as sparse, dynamic graphs. These dynamic graphs can then be fed into a GNN as sequences of timed events to produce feature dictionaries that are resilient to environmental uncertainty and agnostic to ping direction. The ability of GNNs to learn complex systems of interactions makes them a great choice for braid-like feature discovery. To handle the evolving dynamic features of the sonar spectra graphs, a variation of a GNN, called temporal graph networks (TGNs), is used. TGNs utilize memory modules and graph-based operators to outperform previous GNN-based approaches when handling dynamic graphs. We use TGNs to model the evolution of the sonar spectra graphs and ultimately perform graph-based classification. Preliminary results performed on the Malta Plateau field experiment are presented.
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He, Ya Feng, Qing Yu, and You Ming Wang. "Topological Synthesis of Planetary Gear Trains." Applied Mechanics and Materials 33 (October 2010): 43–50. http://dx.doi.org/10.4028/www.scientific.net/amm.33.43.
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Application of topological graphs to analyze and synthesize planetary gear trains (PGTs) is an effective approach in searching for innovative planetary trains. Two topological graphs of basic structure and three fundamental units of PGTs are established as the basis of topological synthesis. Then combination system of PGTs are summarized based on stratify-cation, which can be classified as four basic type including series, parallel, separate and close. Several kinds of compound combination are also discussed. The variation law of topological graphs are analyzed and sorted as diversification including evolution, regression, decay and recovery and inversion transforming graphs to PGTs. Finally graph library based on variation law is presented as a simple and practical way to create a great number of new PGTs.
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Aldous, David. "Hitting times for random walks on vertex-transitive graphs." Mathematical Proceedings of the Cambridge Philosophical Society 106, no. 1 (July 1989): 179–91. http://dx.doi.org/10.1017/s0305004100068079.
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AbstractFor random walks on finite graphs, we record some equalities, inequalities and limit theorems (as the size of graph tends to infinity) which hold for vertex-transitive graphs but not for general regular graphs. The main result is a sharp condition for asymptotic exponentiality of the hitting time to a single vertex. Another result is a lower bound for the coefficient of variation of hitting times. Proofs exploit the complete monotonicity properties of the associated continuous-time walk.
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Gomez-Pilar, Javier, Jesús Poza, Alejandro Bachiller, Carlos Gómez, Pablo Núñez, Alba Lubeiro, Vicente Molina, and Roberto Hornero. "Quantification of Graph Complexity Based on the Edge Weight Distribution Balance: Application to Brain Networks." International Journal of Neural Systems 28, no. 01 (December 20, 2017): 1750032. http://dx.doi.org/10.1142/s0129065717500320.
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The aim of this study was to introduce a novel global measure of graph complexity: Shannon graph complexity (SGC). This measure was specifically developed for weighted graphs, but it can also be applied to binary graphs. The proposed complexity measure was designed to capture the interplay between two properties of a system: the ‘information’ (calculated by means of Shannon entropy) and the ‘order’ of the system (estimated by means of a disequilibrium measure). SGC is based on the concept that complex graphs should maintain an equilibrium between the aforementioned two properties, which can be measured by means of the edge weight distribution. In this study, SGC was assessed using four synthetic graph datasets and a real dataset, formed by electroencephalographic (EEG) recordings from controls and schizophrenia patients. SGC was compared with graph density (GD), a classical measure used to evaluate graph complexity. Our results showed that SGC is invariant with respect to GD and independent of node degree distribution. Furthermore, its variation with graph size [Formula: see text] is close to zero for [Formula: see text]. Results from the real dataset showed an increment in the weight distribution balance during the cognitive processing for both controls and schizophrenia patients, although these changes are more relevant for controls. Our findings revealed that SGC does not need a comparison with null-hypothesis networks constructed by a surrogate process. In addition, SGC results on the real dataset suggest that schizophrenia is associated with a deficit in the brain dynamic reorganization related to secondary pathways of the brain network.
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Mahmood, Faisal, Nauman Shahid, Ulf Skoglund, and Pierre Vandergheynst. "Adaptive Graph-Based Total Variation for Tomographic Reconstructions." IEEE Signal Processing Letters 25, no. 5 (May 2018): 700–704. http://dx.doi.org/10.1109/lsp.2018.2816582.
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MANUEL, PAUL, and SANDI KLAVŽAR. "A GENERAL POSITION PROBLEM IN GRAPH THEORY." Bulletin of the Australian Mathematical Society 98, no. 2 (July 18, 2018): 177–87. http://dx.doi.org/10.1017/s0004972718000473.
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The paper introduces a graph theory variation of the general position problem: given a graph $G$, determine a largest set $S$ of vertices of $G$ such that no three vertices of $S$ lie on a common geodesic. Such a set is a max-gp-set of $G$ and its size is the gp-number $\text{gp}(G)$ of $G$. Upper bounds on $\text{gp}(G)$ in terms of different isometric covers are given and used to determine the gp-number of several classes of graphs. Connections between general position sets and packings are investigated and used to give lower bounds on the gp-number. It is also proved that the general position problem is NP-complete.
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Anshar, Muhammad, Raden Sumiharto, and Moh Edi Wibowo. "Progressive Content Generation Based on Cyclic Graph for Generate Dungeon." IJCCS (Indonesian Journal of Computing and Cybernetics Systems) 17, no. 1 (February 23, 2023): 91. http://dx.doi.org/10.22146/ijccs.81178.
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Dungeon is level in game consisting collection of rooms and doors with obstacles inside. To make good level, takes a lot of time. With Procedural Content Generation (PCG), dungeons can be created automatically. One of the approaches in PCG to create levels is progressive. Progressive approach produces timeline as representation of the interactions in the game. Timeline representation that is in the form of one straight line is good for endless runner, but for dungeon, the levels are linear. In this research, the timeline is changed to cyclic graph. Cyclic graph is formed using graph grammar algorithm. This research aims to build dungeon that has not linear and minimal dead ends. To eliminate linearity in dungeons, branching in dungeons needs to be formed. The steps carried out in this research are designing graph grammar rules, generating population of graphs, evaluating graphs with fitness values, and building dungeons. Four functions are used to determine the fitness value: shortest vertices, average duration, replayability, and variation. Dungeons produced with progressive approach manage to minimize linearity in dungeons. Dungeon formation is very dependent on the rule grammar that forms it. With the evaluation process, linear dungeons resulting from grammar rules can be minimized.
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GEUVERS, HERMAN, and IRIS LOEB. "Natural deduction via graphs: formal definition and computation rules." Mathematical Structures in Computer Science 17, no. 3 (June 2007): 485–526. http://dx.doi.org/10.1017/s0960129507006123.
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In this paper, we introduce the formalism of deduction graphs as a generalisation of both Gentzen–Prawitz style natural deduction and Fitch style flag deduction. The advantage of this formalism is that, as with flag deductions (but not natural deduction), subproofs can be shared, but the linearisation used in flag deductions is avoided. Our deduction graphs have both nodes and boxes, which are collections of nodes that also form a node themselves. This is reminiscent of the bigraphs of Milner, where the link graph describes the nodes and edges and the place graph describes the nesting of nodes. We give a precise definition of deduction graphs, together with some illustrative examples. Furthermore, we analyse their computational behaviour by studying the process of cut-elimination and by defining translations from deduction graphs to simply typed lambda terms. From a slight variation of this translation, we conclude that the process of cut-elimination is strongly normalising. The translation to simple type theory removes quite a lot of structure, so we also propose a translation to a context calculus with lets that faithfully captures the structure of deduction graphs. The proof nets of linear logic also offer a graph-like presentation of natural deduction, and we point out some similarities between the two formalisms.
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Ghaffaari, Ali, and Tobias Marschall. "Fully-sensitive seed finding in sequence graphs using a hybrid index." Bioinformatics 35, no. 14 (July 2019): i81—i89. http://dx.doi.org/10.1093/bioinformatics/btz341.
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Abstract Motivation Sequence graphs are versatile data structures that are, for instance, able to represent the genetic variation found in a population and to facilitate genome assembly. Read mapping to sequence graphs constitutes an important step for many applications and is usually done by first finding exact seed matches, which are then extended by alignment. Existing methods for finding seed hits prune the graph in complex regions, leading to a loss of information especially in highly polymorphic regions of the genome. While such complex graph structures can indeed lead to a combinatorial explosion of possible alleles, the query set of reads from a diploid individual realizes only two alleles per locus—a property that is not exploited by extant methods. Results We present the Pan-genome Seed Index (PSI), a fully-sensitive hybrid method for seed finding, which takes full advantage of this property by combining an index over selected paths in the graph with an index over the query reads. This enables PSI to find all seeds while eliminating the need to prune the graph. We demonstrate its performance with different parameter settings on both simulated data and on a whole human genome graph constructed from variants in the 1000 Genome Project dataset. On this graph, PSI outperforms GCSA2 in terms of index size, query time and sensitivity. Availability and implementation The C++ implementation is publicly available at: https://github.com/cartoonist/psi.
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Iršič, Vesna, and Sandi Klavžar. "Strong geodetic problem on Cartesian products of graphs." RAIRO - Operations Research 52, no. 1 (January 2018): 205–16. http://dx.doi.org/10.1051/ro/2018003.
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The strong geodetic problem is a recent variation of the geodetic problem. For a graph G, its strong geodetic number sg(G) is the cardinality of a smallest vertex subset S, such that each vertex of G lies on a fixed shortest path between a pair of vertices from S. In this paper, the strong geodetic problem is studied on the Cartesian product of graphs. A general upper bound for sg(G □ H) is determined, as well as exact values for Km □ Kn, K1,k □ Pl, and prisms over Kn–e. Connections between the strong geodetic number of a graph and its subgraphs are also discussed.
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Carswell, C. Melody, and Cathy Emery. "Effects of Stimulus Complexity and Cognitive Style on Spontaneous Interpretations of Line Graphs." Proceedings of the Human Factors Society Annual Meeting 36, no. 18 (October 1992): 1498–502. http://dx.doi.org/10.1177/154193129203601829.
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Thirty adult subjects studied each of eighteen single-function line graphs for self-determined periods. The structural complexity of the stimulus graphs was varied in three ways: through addition of data points, reversal of trends, and elimination of symmetry. Subjects provided written interpretations immediately following examination of each graph. Indirect indices of comprehensibility (i.e., increased graph study times and increased content in the written interpretations) suggested that trend reversals were the primary determinant of complexity. While the number of data points and the presence or absence of symmetry were not associated with longer study times or greater overall content production, varying these structural features did lead to strategic shifts in the interpretive emphasis on global versus local features of the graphically-displayed data. Specifically, the presence of symmetry or the addition of data points led to increases in global content and decreases in local content. Lastly, cognitive style of subjects was systematically related to graphical interpretation. Impulsive subjects were less likely than reflective subjects to interpret local features of the graph, and were also less sensitive to variation in structural characteristics.
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Garrison, Erik, Jouni Sirén, Adam M. Novak, Glenn Hickey, Jordan M. Eizenga, Eric T. Dawson, William Jones, et al. "Variation graph toolkit improves read mapping by representing genetic variation in the reference." Nature Biotechnology 36, no. 9 (October 2018): 875–79. http://dx.doi.org/10.1038/nbt.4227.
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Sudbø, J., R. Marcelpoil, and A. Reith. "Caveats: Numerical Requirements in Graph Theory Based Quantitation of Tissue Architecture." Analytical Cellular Pathology 21, no. 2 (2000): 59–69. http://dx.doi.org/10.1155/2000/438202.
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Graph theory based methods represent one approach to an objective and reproducible structural analysis of tissue architecture. By these methods, neighborhood relations between a number of objects (e.g., cells) are explored and inherent to these methods are therefore certain requirements as to the number of objects to be included in the analysis. However, the question of how many objects are required to achieve reproducible values in repeated computations of proposed structural features, has previously not been adressed specifically.After digitising HE stained slides and storing them as grey level images, cell nuclei were segmented and their geometrical centre of gravity were computed, serving as the basis for construction of the Voronoi diagram (VD) and its subgraphs. Variations in repeated computations of structural features derived from these graphs were related to the number of cell nuclei included in the analysis.We demonstrate a large variation in the values of the structural features from one computation to another in one and the same section when only a limited number of cells (100–500) are included in the analysis. This variation decreased with increasing number of cells analyzed. The exact number of cells required to achieve reproducible values differ significantly between tissues, but not between separate cases of similar lesions. There are no significant differences between normal and malignantly changed tissues in oral mucosa with respect to how many cells must be included.For graph theory based analysis of tissue architecture, care must be taken to include an adequate number of objects; for some of the structural features we have tested, more than 3000 cells.
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Umilasari, Reni, Ilham Saifudin, and Isnawati Lujeng Lestari. "Star metric dimension of complete, bipartite, complete bipartite and fan graphs." International Journal of Trends in Mathematics Education Research 5, no. 2 (June 30, 2022): 199–205. http://dx.doi.org/10.33122/ijtmer.v5i2.137.
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One of the topics in graph theory that is interesting and developed continuously is metric dimension. It has some new variation concepts, such as star metric dimension. The order set of Z={z1,z2,…,zn }⊆V(G) called star resolving set of connected graph G if Z is Star Graph and for every vertex in G has different representation to the set Z. The representation is expressed as the distance d(u,z), it is the shortest path from vertex u to z for every u,z∈V(G). Star basis of a graph is the smallest cardinality of star resolving set. The number of vertex in star basis is called star metric dimension of G which denoted by Sdim(G). The purpose of this article is to determine the characteristic of star metric dimension and the value of star metric dimension of some classes of graphs. The method which is used in this study is library research. Some of the results of this research are complete graph has Sdim(Kn )=n-1, for n≥3, bipartite graph K(2,n) has Sdim (K(2,n) )=n, for n≥3. Besides complete bipartite graph hasn’t star metric dimension or for m,n≥3 or it can said that Sdim (K(m,n) )=0. Another graph, that is Fan graph has Sdim (Fn )=2 for 2≤n≤5 and for n≥6 Sdim (Fn )=(2n+3)/5.
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Blinova, I. V., A. S. Gnedash, and I. Y. Popov. "A time-dependent metric graph with a fourth-order operator on the edges." Theoretical and Applied Mechanics 48, no. 2 (2021): 187–200. http://dx.doi.org/10.2298/tam200928007b.
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The metric graph model is suggested for description of elastic vibration in a network of rods under the assumption that the rod lengths vary in time. A single rod and star-like graph are considered. Influence of the length variation law on the vibration distribution is investigated. For high-frequency length variation one observes a fast transition to high-frequency amplitude distribution.
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van Gennip, Yves. "An MBO Scheme for Minimizing the Graph Ohta–Kawasaki Functional." Journal of Nonlinear Science 30, no. 5 (June 1, 2018): 2325–73. http://dx.doi.org/10.1007/s00332-018-9468-8.
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Abstract We study a graph-based version of the Ohta–Kawasaki functional, which was originally introduced in a continuum setting to model pattern formation in diblock copolymer melts and has been studied extensively as a paradigmatic example of a variational model for pattern formation. Graph-based problems inspired by partial differential equations (PDEs) and variational methods have been the subject of many recent papers in the mathematical literature, because of their applications in areas such as image processing and data classification. This paper extends the area of PDE inspired graph-based problems to pattern-forming models, while continuing in the tradition of recent papers in the field. We introduce a mass conserving Merriman–Bence–Osher (MBO) scheme for minimizing the graph Ohta–Kawasaki functional with a mass constraint. We present three main results: (1) the Lyapunov functionals associated with this MBO scheme $$\Gamma $$ Γ -converge to the Ohta–Kawasaki functional (which includes the standard graph-based MBO scheme and total variation as a special case); (2) there is a class of graphs on which the Ohta–Kawasaki MBO scheme corresponds to a standard MBO scheme on a transformed graph and for which generalized comparison principles hold; (3) this MBO scheme allows for the numerical computation of (approximate) minimizers of the graph Ohta–Kawasaki functional with a mass constraint.
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HAMID, I. SAHUL, and MAYAMMA JOSEPH. "FURTHER RESULTS ON INDUCED GRAPHOIDAL DECOMPOSITION." Discrete Mathematics, Algorithms and Applications 05, no. 01 (March 2013): 1350006. http://dx.doi.org/10.1142/s1793830913500067.
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Let G be a nontrivial, simple, finite, connected and undirected graph. A graphoidal decomposition (GD) of G is a collection ψ of paths and cycles in G that are internally disjoint such that every edge of G lies in exactly one member of ψ. As a variation of GD the notion of induced graphoidal decomposition (IGD) was introduced in [S. Arumugam, Path covers in graphs (2006)] which is a GD all of whose members are either induced paths or induced cycles. The minimum number of elements in such a decomposition of a graph G is called the IGD number, denoted by ηi(G). In this paper, we extend the study of the parameter ηi by establishing bounds for ηi(G) in terms of the diameter, girth and the maximum degree along with characterization of graphs achieving the bounds.