Journal articles on the topic 'EDGE TEST TREE GRAPH'
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Wei, Yuxuan, Zhinan Gao, and Xingyan Lu. "The Complexity of Wheel Graphs with Multiple Edges and Vertices." Asian Research Journal of Mathematics 19, no. 9 (June 19, 2023): 1–12. http://dx.doi.org/10.9734/arjom/2023/v19i9694.
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In this paper, we focus on calculate the number of spanning trees of the general wheel graphs, which meansthe original wheel graphs adding large amount of vertices and edges. Particularly, we introduce the C-graphand deduce a new equation that computing the spanning trees by removing C-graphs instead of edges.In Addition, we test our results by Kirchhoff’s matrix-tree theorem in some simple cases and provide thetree entropy of the general wheel graphs. Finally, we analyse the relation between the wheel graph anddouble-wheel graphs and propose the idea of calculating the spanning trees of double-wheel graphs.
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Wu, Yuezhong, Qiang Liu, Rongrong Chen, Changyun Li, and Ziran Peng. "A Group Recommendation System of Network Document Resource Based on Knowledge Graph and LSTM in Edge Computing." Security and Communication Networks 2020 (December 2, 2020): 1–11. http://dx.doi.org/10.1155/2020/8843803.
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The Internet has become one of the important channels for users to obtain information and knowledge. It is crucial to work out how to acquire personalized requirement of users accurately and effectively from huge amount of network document resources. Group recommendation is an information system for group participation in common activities that meets the common interests of all members in the group. This paper proposes a group recommendation system for network document resource exploration using the knowledge graph and LSTM in edge computing, which can solve the problem of information overload and resource trek effectively. An extensive system test has been carried out in the field of big data application in packaging industry. The experimental results show that the proposed system recommends network document resource more accurately and further improves recommendation quality using the knowledge graph and LSTM in edge computing. Therefore, it can meet the user’s personalized resource need more effectively.
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Zhong, Shuaihao, Duoqiang Wang, Wei Li, Feng Lu, and Hai Jin. "Burner: Recipe Automatic Generation for HPC Container Based on Domain Knowledge Graph." Wireless Communications and Mobile Computing 2022 (May 25, 2022): 1–14. http://dx.doi.org/10.1155/2022/4592428.
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As one of the emerging cloud computing technologies, containers are widely used in academia and industry. The cloud computing built by the container in the high performance computing (HPC) center can provide high-quality services to users at the edge. Singularity Definition File and Dockerfile (we refer to such files as recipes) have attracted wide attention due to their encapsulation of the application running environment in a container. However, creating a recipe requires extensive domain knowledge, which is error-prone and time-consuming. Accordingly, more than 34% of Dockerfiles in Github cannot successfully build container images. The crucial points about recipe creation include selecting the entities (base images and packages) and determining their relationships (correct installation order for transitive dependencies). Since the relationships between entities can be expressed accurately and efficiently by the knowledge graph, we introduce knowledge graph to generate high-quality recipes automatically. This paper proposes an automatic recipe generation system named Burner, enabling users with no professional computer background to generate the recipes. We first develop a toolset including a recipe parser and an entity-relationship miner. Our two-phase recipe parsing method can perform abstract syntax tree (AST) parsing more deeply on the recipe file to achieve entity extraction; the parsing success rate (PSR) of the two-phase parsing method is 10.1% higher than the one-phase parsing. Then, we build a knowledge base containing 2,832 entities and 62,614 entity relationships, meeting the needs of typical HPC applications. In the test of image build, the singularity image build success rate reaches 80%. Compared with the ItemCF recommendation method, our recommendation method TB-TFIDF achieves a performance improvement by up to 50.86%.
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Gilani, S. A. N., M. Awrangjeb, and G. Lu. "FUSION OF LIDAR DATA AND MULTISPECTRAL IMAGERY FOR EFFECTIVE BUILDING DETECTION BASED ON GRAPH AND CONNECTED COMPONENT ANALYSIS." ISPRS - International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences XL-3/W2 (March 10, 2015): 65–72. http://dx.doi.org/10.5194/isprsarchives-xl-3-w2-65-2015.
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Building detection in complex scenes is a non-trivial exercise due to building shape variability, irregular terrain, shadows, and occlusion by highly dense vegetation. In this research, we present a graph based algorithm, which combines multispectral imagery and airborne LiDAR information to completely delineate the building boundaries in urban and densely vegetated area. In the first phase, LiDAR data is divided into two groups: ground and non-ground data, using ground height from a bare-earth DEM. A mask, known as the primary building mask, is generated from the non-ground LiDAR points where the black region represents the elevated area (buildings and trees), while the white region describes the ground (earth). The second phase begins with the process of Connected Component Analysis (CCA) where the number of objects present in the test scene are identified followed by initial boundary detection and labelling. Additionally, a graph from the connected components is generated, where each black pixel corresponds to a node. An edge of a unit distance is defined between a black pixel and a neighbouring black pixel, if any. An edge does not exist from a black pixel to a neighbouring white pixel, if any. This phenomenon produces a disconnected components graph, where each component represents a prospective building or a dense vegetation (a contiguous block of black pixels from the primary mask). In the third phase, a clustering process clusters the segmented lines, extracted from multispectral imagery, around the graph components, if possible. In the fourth step, NDVI, image entropy, and LiDAR data are utilised to discriminate between vegetation, buildings, and isolated building’s occluded parts. Finally, the initially extracted building boundary is extended pixel-wise using NDVI, entropy, and LiDAR data to completely delineate the building and to maximise the boundary reach towards building edges. The proposed technique is evaluated using two Australian data sets: Aitkenvale and Hervey Bay, for object-based and pixel-based completeness, correctness, and quality. The proposed technique detects buildings larger than 50 m<sup>2</sup> and 10 m<sup>2</sup> in the Aitkenvale site with 100% and 91% accuracy, respectively, while in the Hervey Bay site it performs better with 100% accuracy for buildings larger than 10 m<sup>2</sup> in area.
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Chimani, Markus, Giuseppe Di Battista, Fabrizio Frati, and Karsten Klein. "Advances on Testing C-Planarity of Embedded Flat Clustered Graphs." International Journal of Foundations of Computer Science 30, no. 02 (February 2019): 197–230. http://dx.doi.org/10.1142/s0129054119500011.
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In this paper, we show a polynomial-time algorithm for testing [Formula: see text]-planarity of embedded flat clustered graphs with at most two vertices per cluster on each face. Our result is based on a reduction to the planar set of spanning trees in topological multigraphs (pssttm) problem, which is defined as follows. Given a (non-planar) topological multigraph [Formula: see text] with [Formula: see text] connected components [Formula: see text], do spanning trees of [Formula: see text] exist such that no two edges in any two spanning trees cross? Kratochvíl et al. [SIAM Journal on Discrete Mathematics, 4(2): 223–244, 1991] proved that the problem is NP-hard even if [Formula: see text]; on the other hand, Di Battista and Frati presented a linear-time algorithm to solve the pssttm problem for the case in which [Formula: see text] is a [Formula: see text]-planar topological multigraph [Journal of Graph Algorithms and Applications, 13(3): 349–378, 2009]. For any embedded flat clustered graph [Formula: see text], an instance [Formula: see text] of the pssttm problem can be constructed in polynomial time such that [Formula: see text] is [Formula: see text]-planar if and only if [Formula: see text] admits a solution. We show that, if [Formula: see text] has at most two vertices per cluster on each face, then it can be tested in polynomial time whether the corresponding instance [Formula: see text] of the pssttm problem is positive or negative. Our strategy for solving the pssttm problem on [Formula: see text] is to repeatedly perform a sequence of tests, which might let us conclude that [Formula: see text] is a negative instance, and simplifications, which might let us simplify [Formula: see text] by removing or contracting some edges. Most of these tests and simplifications are performed “locally”, by looking at the crossings involving a single edge or face of a connected component [Formula: see text] of [Formula: see text]; however, some tests and simplifications have to consider certain global structures in [Formula: see text], which we call [Formula: see text]-donuts. If no test concludes that [Formula: see text] is a negative instance of the pssttm problem, then the simplifications eventually transform [Formula: see text] into an equivalent [Formula: see text]-planar topological multigraph on which we can apply the cited linear-time algorithm by Di Battista and Frati.
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Abbasi, Mozhgan, Jochem Verrelst, Mohsen Mirzaei, Safar Marofi, and Hamid Reza Riyahi Bakhtiari. "Optimal Spectral Wavelengths for Discriminating Orchard Species Using Multivariate Statistical Techniques." Remote Sensing 12, no. 1 (December 23, 2019): 63. http://dx.doi.org/10.3390/rs12010063.
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Sustainable management of orchard fields requires detailed information about the tree types, which is a main component of precision agriculture programs. To this end, hyperspectral imagery can play a major role in orchard tree species mapping. Efficient use of hyperspectral data in combination with field measurements requires the development of optimized band selection strategies to separate tree species. In this study, field spectroscopy (350 to 2500 nm) was performed through scanning 165 spectral leaf samples of dominant orchard tree species (almond, walnut, and grape) in Chaharmahal va Bakhtiyari province, Iran. Two multivariable methods were employed to identify the optimum wavelengths: the first includes three-step approach ANOVA, random forest classifier (RFC) and principal component analysis (PCA), and the second employs partial least squares (PLS). For both methods we determined whether tree species can be spectrally separated using discriminant analysis (DA) and then the optimal wavelengths were identified for this purpose. Results indicate that all species express distinct spectral behaviors at the beginning of the visible range (from 350 to 439 nm), the red edge and the near infrared wavelengths (from 701 to 1405 nm). The ANOVA test was able to reduce primary wavelengths (2151) to 792, which had a significant difference (99% confidence level), then the RFC further reduced the wavelengths to 118. By removing the overlapping wavelengths, the PCA represented five components (99.87% of variance) which extracted optimal wavelengths were: 363, 423, 721, 1064, and 1388 nm. The optimal wavelengths for the species discrimination using the best PLS-DA model (100% accuracy) were at 397, 515, 647, 1386, and 1919 nm.
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Raghavan, S., and Rui Zhang. "Influence Maximization with Latency Requirements on Social Networks." INFORMS Journal on Computing 34, no. 2 (March 2022): 710–28. http://dx.doi.org/10.1287/ijoc.2021.1095.
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Targeted marketing strategies are of significant interest in the smartapp economy. Typically, one seeks to identify individuals to strategically target in a social network so that the network is influenced at a minimal cost. In many practical settings, the effects of direct influence predominate, leading to the positive influence dominating set with partial payments (PIDS-PP) problem that we discuss in this paper. The PIDS-PP problem is NP-complete because it generalizes the dominating set problem. We discuss several mixed integer programming formulations for the PIDS-PP problem. First, we describe two compact formulations on the payment space. We then develop a stronger compact extended formulation. We show that when the underlying graph is a tree, this compact extended formulation provides integral solutions for the node selection variables. In conjunction, we describe a polynomial-time dynamic programming algorithm for the PIDS-PP problem on trees. We project the compact extended formulation onto the payment space, providing an equivalently strong formulation that has exponentially many constraints. We present a polynomial time algorithm to solve the associated separation problem. Our computational experience on a test bed of 100 real-world graph instances (with up to approximately 465,000 nodes and 835,000 edges) demonstrates the efficacy of our strongest payment space formulation. It finds solutions that are on average 0.4% from optimality and solves 80 of the 100 instances to optimality. Summary of Contribution: The study of influence propagation is important in a number of applications including marketing, epidemiology, and healthcare. Typically, in these problems, one seeks to identify individuals to strategically target in a social network so that the entire network is influenced at a minimal cost. With the ease of tracking consumers in the smartapp economy, the scope and nature of these problems have become larger. Consequently, there is considerable interest across multiple research communities in computationally solving large-scale influence maximization problems, which thus represent significant opportunities for the development of operations research–based methods and analysis in this interface. This paper introduces the positive influence dominating set with partial payments (PIDS-PP) problem, an influence maximization problem where the effects of direct influence predominate, and it is possible to make partial payments to nodes that are not targeted. The paper focuses on model development to solve large-scale PIDS-PP problems. To this end, starting from an initial base optimization model, it uses several operations research model strengthening techniques to develop two equivalent models that have strong computational performance (and can be theoretically shown to be the best model for trees). Computational experiments on a test bed of 100 real-world graph instances (with up to approximately 465,000 nodes and 835,000 edges) attest to the efficacy of the best model, which finds solutions that are on average 0.4% from optimality and solves 80 of the 100 instances to optimality.
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Zhang, Zhijun, Muhammad Awais Umar, Xiaojun Ren, Basharat Rehman Ali, Mujtaba Hussain, and Xiangmei Li. "Tree-Antimagicness of Web Graphs and Their Disjoint Union." Mathematical Problems in Engineering 2020 (April 9, 2020): 1–6. http://dx.doi.org/10.1155/2020/4565829.
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In graph theory, the graph labeling is the assignment of labels (represented by integers) to edges and/or vertices of a graph. For a graph G=V,E, with vertex set V and edge set E, a function from V to a set of labels is called a vertex labeling of a graph, and the graph with such a function defined is called a vertex-labeled graph. Similarly, an edge labeling is a function of E to a set of labels, and in this case, the graph is called an edge-labeled graph. In this research article, we focused on studying super ad,d-T4,2-antimagic labeling of web graphs W2,n and isomorphic copies of their disjoint union.
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Haghir Chehreghani, Morteza. "Unsupervised representation learning with Minimax distance measures." Machine Learning 109, no. 11 (July 28, 2020): 2063–97. http://dx.doi.org/10.1007/s10994-020-05886-4.
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Abstract We investigate the use of Minimax distances to extract in a nonparametric way the features that capture the unknown underlying patterns and structures in the data. We develop a general-purpose and computationally efficient framework to employ Minimax distances with many machine learning methods that perform on numerical data. We study both computing the pairwise Minimax distances for all pairs of objects and as well as computing the Minimax distances of all the objects to/from a fixed (test) object. We first efficiently compute the pairwise Minimax distances between the objects, using the equivalence of Minimax distances over a graph and over a minimum spanning tree constructed on that. Then, we perform an embedding of the pairwise Minimax distances into a new vector space, such that their squared Euclidean distances in the new space equal to the pairwise Minimax distances in the original space. We also study the case of having multiple pairwise Minimax matrices, instead of a single one. Thereby, we propose an embedding via first summing up the centered matrices and then performing an eigenvalue decomposition to obtain the relevant features. In the following, we study computing Minimax distances from a fixed (test) object which can be used for instance in K-nearest neighbor search. Similar to the case of all-pair pairwise Minimax distances, we develop an efficient and general-purpose algorithm that is applicable with any arbitrary base distance measure. Moreover, we investigate in detail the edges selected by the Minimax distances and thereby explore the ability of Minimax distances in detecting outlier objects. Finally, for each setting, we perform several experiments to demonstrate the effectiveness of our framework.
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Toyonaga, Kenji. "The location of classified edges due to the change in the geometric multiplicity of an eigenvalue in a tree." Special Matrices 7, no. 1 (January 1, 2019): 257–62. http://dx.doi.org/10.1515/spma-2019-0019.
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Abstract Given a combinatorially symmetric matrix A whose graph is a tree T and its eigenvalues, edges in T can be classified in four categories, based upon the change in geometric multiplicity of a particular eigenvalue, when the edge is removed. We investigate a necessary and sufficient condition for each classification of edges. We have similar results as the case for real symmetric matrices whose graph is a tree. We show that a g-2-Parter edge, a g-Parter edge and a g-downer edge are located separately from each other in a tree, and there is a g-neutral edge between them. Furthermore, we show that the distance between a g-downer edge and a g-2-Parter edge or a g-Parter edge is at least 2 in a tree. Lastly we give a combinatorially symmetric matrix whose graph contains all types of edges.
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Mary, Francis Remigius Perpetua, Swaminathan Mohanaselvi, and Said Broumi. "A solution approach to minimum spanning tree problem under fermatean fuzzy environment." Bulletin of Electrical Engineering and Informatics 12, no. 3 (June 1, 2023): 1738–46. http://dx.doi.org/10.11591/eei.v12i3.4794.
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In classical graph theory, the minimal spanning tree (MST) is a subgraph with no cycles that connects each vertex with minimum edge weights. Calculating minimum spanning tree of a graph has always been a common problem throughout ages. Fuzzy minimum spanning tree (FMST) is able to handle uncertainty existing in edge weights for a fuzzy graph which occurs in real world situations. In this article, we have studied the MST problem of a directed and undirected fuzzy graph whose edge weights are represented by fermatean fuzzy numbers (FFN). We focus on determining an algorithmic approach for solving fermatean fuzzy minimum spanning tree (FFMST) using the modified Prim’s algorithm for an undirected graph and modified optimum branching algorithm for a directed graph under FFN environment. Since the proposed algorithm includes FFN ranking and arithmetic operations, we use FFNs improved scoring function to compare the weights of the edges of the graph. With the help of numerical examples, the solution technique for the proposed FFMST model is described.
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Smirnov, Alexander V. "The Spanning Tree of a Divisible Multiple Graph." Modeling and Analysis of Information Systems 25, no. 4 (August 27, 2018): 388–401. http://dx.doi.org/10.18255/1818-1015-2018-4-388-401.
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In this paper, we study undirected multiple graphs of any natural multiplicity k > 1. There are edges of three types: ordinary edges, multiple edges and multi-edges. Each edge of the last two types is a union of k linked edges, which connect 2 or k + 1 vertices, correspondingly. The linked edges should be used simultaneously. If a vertex is incident to a multiple edge, it can be also incident to other multiple edges, and it can be the common ending vertex to k linked edges of a multi-edge. If a vertex is the common end of some multi-edge, it cannot be the common end of any other multi-edge. Special attention is paid to the class of divisible multiple graphs. The main peculiarity of them is a possibility to divide the graph into k parts, which are adjusted on the linked edges and which have no common edges. Each part is an ordinary graph. The definition of a multiple tree is stated and the basic properties of such trees are studied. Unlike ordinary trees, the number of edges in a multiple tree is not fixed. In the article, the evaluation of the minimum and maximum number of edges in the divisible tree is stated and proved. Next, the definitions of the spanning tree and the complete spanning tree of a multiple graph are given. The criterion of completeness of the spanning tree is proved for divisible graphs. It is also proved that a complete spanning tree exists in any divisible graph. If the multiple graph is weighted, the minimum spanning tree problem and the minimum complete spanning tree problem can be set. In the article, we suggest a heuristic algorithm for the minimum complete spanning tree problem for a divisible graph.
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Prasad, K. C. Rajendra, Venkanagouda M. Goudar, and K. M. Niranjan. "Pathos edge semi-middle graph of a tree." Malaya Journal of Matematik 8, no. 4 (2020): 2190–93. http://dx.doi.org/10.26637/mjm0804/0148.
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Triyani, Triyani, and Irham Taufiq. "BEBERAPA SIFAT HIMPUNAN KRITIS PADA PELABELAN AJAIB GRAF BANANA TREE." Jurnal Ilmiah Matematika dan Pendidikan Matematika 4, no. 2 (December 28, 2012): 271. http://dx.doi.org/10.20884/1.jmp.2012.4.2.2963.
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A critical set in edge magic total labeling on graph G is a subset label such that it can forms the edge magic total labeling uniquely. This paper investigate critical set on Banana Tree graph. The result shows some properties of critical set on Banana Tree graph, such as the size of it at least , where n is number of leaf and k is number of star, except the size of critical set on graph BT(1,1) is 2. Beside it, if x is the label of any leaf and y is the label of the edge adjacent to it then each critical set in λ must contain either x or y, not both.
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Smirnov, Alexander Valeryevich. "NP-completeness of the Minimum Spanning Tree Problem of a Multiple Graph of Multiplicity k ≥ 3." Modeling and Analysis of Information Systems 28, no. 1 (March 24, 2021): 22–37. http://dx.doi.org/10.18255/1818-1015-2021-1-22-37.
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In this paper, we study undirected multiple graphs of any natural multiplicity k > 1. There are edges of three types: ordinary edges, multiple edges and multi-edges. Each edge of the last two types is a union of k linked edges, which connect 2 or (k + 1) vertices correspondingly. The linked edges should be used simultaneously. If a vertex is incident to a multiple edge, it can be also incident to other multiple edges and it can be the common end of k linked edges of some multi-edge. If a vertex is the common end of some multi-edge, it cannot be the common end of another multi-edge. A multiple tree is a connected multiple graph with no cycles. Unlike ordinary trees, the number of edges in a multiple tree is not fixed. The problem of finding the spanning tree can be set for a multiple graph. Complete spanning trees form a special class of spanning trees of a multiple graph. Their peculiarity is that a multiple path joining any two selected vertices exists in the tree if and only if such a path exists in the initial graph. If the multiple graph is weighted, the minimum spanning tree problem and the minimum complete spanning tree problem can be set. Also we can formulate the problems of recognition of the spanning tree and complete spanning tree of the limited weight. The main result of this article is the proof of NPcompleteness of such recognition problems for arbitrary multiple graphs as well as for divisible multiple graphs in the case when multiplicity k ≥ 3. The corresponding optimization problems are NP-hard.
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Jr., Isagani S. Cabahug,. "On Spanning Tree Packing Number of the Complement of Generalized Petersen Graph and Cocktail Party Graph." Asian Research Journal of Mathematics 19, no. 9 (July 21, 2023): 226–32. http://dx.doi.org/10.9734/arjom/2023/v19i9714.
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For any graph G, the spanning tree packing number of \(\sigma\) (G), is the maximum number of edge-disjoint spanning trees contained in G. In this study, we determined the maximum number of edge-disjoint spanning trees of the generalized petersen graph and cocktail graph.
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Bianchi, Maria Paola, Hans-Joachim Böckenhauer, Tatjana Brülisauer, Dennis Komm, and Beatrice Palano. "Online Minimum Spanning Tree with Advice." International Journal of Foundations of Computer Science 29, no. 04 (June 2018): 505–27. http://dx.doi.org/10.1142/s0129054118410034.
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In the online minimum spanning tree problem, a graph is revealed vertex by vertex; together with every vertex, all edges to vertices that are already known are given, and an online algorithm must irrevocably choose a subset of them as a part of its solution. The advice complexity of an online problem is a means to quantify the information that needs to be extracted from the input to achieve good results. For a graph of size [Formula: see text], we show an asymptotically tight bound of [Formula: see text] on the number of advice bits to produce an optimal solution for any given graph. For particular graph classes, e.g., with bounded degree or a restricted edge weight function, we prove that the upper bound can be drastically reduced; e.g., [Formula: see text] advice bits allow to compute an optimal result if the weight function equals the Euclidean distance; if the graph is complete and has two different edge weights, even a logarithmic number suffices. Some of these results make use of the optimality of Kruskal’s algorithm for the offline setting. We also study the trade-off between the number of advice bits and the achievable competitive ratio. To this end, we perform a reduction from another online problem to obtain a linear lower bound on the advice complexity for any near-optimal solution. Using our results finally allows us to give a lower bound on the expected competitive ratio of any randomized online algorithm for the problem, even on graphs with three different edge weights.
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Rajasekaran, Sanguthevar. "On the Euclidean Minimum Spanning Tree Problem." Computing Letters 1, no. 1 (March 6, 2005): 11–14. http://dx.doi.org/10.1163/1574040053326325.
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Given a weighted graph G(V;E), a minimum spanning tree for G can be obtained in linear time using a randomized algorithm or nearly linear time using a deterministic algorithm. Given n points in the plane, we can construct a graph with these points as nodes and an edge between every pair of nodes. The weight on any edge is the Euclidean distance between the two points. Finding a minimum spanning tree for this graph is known as the Euclidean minimum spanning tree problem (EMSTP). The minimum spanning tree algorithms alluded to before will run in time O(n2) (or nearly O(n2)) on this graph. In this note we point out that it is possible to devise simple algorithms for EMSTP in k- dimensions (for any constant k) whose expected run time is O(n), under the assumption that the points are uniformly distributed in the space of interest.CR Categories: F2.2 Nonnumerical Algorithms and Problems; G.3 Probabilistic Algorithms
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Yadav, RN. "Signed graphs connected with the root lattice." BIBECHANA 11 (May 10, 2014): 157–60. http://dx.doi.org/10.3126/bibechana.v11i0.10396.
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For any base of the root lattice (An) we can construct a signed graph. A signed graph is one whose edges are signed by +1 or -1. A signed graph is balanced if and only if its vertex set can be divided into two sets-either of which may be empty–so that each edge between the sets is negative and each edge within a set is positive. For a given signed graph Tsaranov, Siedel and Cameron constructed the corresponding root lattice. In the present work we have dealt with signed graphs corresponding to the root lattice An. A connected graph is called a Fushimi tree if its all blocks are complete subgraphs. A Fushimi tree is said to be simple when by deleting any cut vertex we have always two connected components. A signed Fushimi tree is called a Fushimi tree with standard sign if it can be transformed into a signed Fushimi tree whose all edges are signed by +1 by switching. Here we have proved that any signed graph corresponding to An is a simple Fushimi tree with standard sign. Our main result is that s simple Fushimi tree with standard sign is contained in the cluster given by a line. DOI: http://dx.doi.org/10.3126/bibechana.v11i0.10396 BIBECHANA 11(1) (2014) 157-160
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Min, Seunghwan, Sung Gwan Park, Kunsoo Park, Dora Giammarresi, Giuseppe F. Italiano, and Wook-Shin Han. "Symmetric continuous subgraph matching with bidirectional dynamic programming." Proceedings of the VLDB Endowment 14, no. 8 (April 2021): 1298–310. http://dx.doi.org/10.14778/3457390.3457395.
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In many real datasets such as social media streams and cyber data sources, graphs change over time through a graph update stream of edge insertions and deletions. Detecting critical patterns in such dynamic graphs plays an important role in various application domains such as fraud detection, cyber security, and recommendation systems for social networks. Given a dynamic data graph and a query graph, the continuous subgraph matching problem is to find all positive matches for each edge insertion and all negative matches for each edge deletion. The state-of-the-art algorithm TurboFlux uses a spanning tree of a query graph for filtering. However, using the spanning tree may have a low pruning power because it does not take into account all edges of the query graph. In this paper, we present a symmetric and much faster algorithm SymBi which maintains an auxiliary data structure based on a directed acyclic graph instead of a spanning tree, which maintains the intermediate results of bidirectional dynamic programming between the query graph and the dynamic graph. Extensive experiments with real and synthetic datasets show that SymBi outperforms the state-of-the-art algorithm by up to three orders of magnitude in terms of the elapsed time.
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Meddah, Nacéra, and Mustapha Chellali. "Edges contained in all or in no minimum edge dominating set of a tree." Discrete Mathematics, Algorithms and Applications 11, no. 04 (August 2019): 1950040. http://dx.doi.org/10.1142/s179383091950040x.
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In a graph, an edge dominates itself and all its adjacent edges. An edge dominating set (EDS) in a graph [Formula: see text] is a subset of edges that dominates every edge of [Formula: see text] In this paper, we characterize edges that are in all or in no minimum EDS in trees.
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Septory, Brian Juned, Liliek Susilowati, Dafik Dafik, and Veerabhadraiah Lokesha. "On the Study of Rainbow Antimagic Connection Number of Comb Product of Friendship Graph and Tree." Symmetry 15, no. 1 (December 21, 2022): 12. http://dx.doi.org/10.3390/sym15010012.
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Given a graph G with vertex set V(G) and edge set E(G), for the bijective function f(V(G))→{1,2,⋯,|V(G)|}, the associated weight of an edge xy∈E(G) under f is w(xy)=f(x)+f(y). If all edges have pairwise distinct weights, the function f is called an edge-antimagic vertex labeling. A path P in the vertex-labeled graph G is said to be a rainbow x−y path if for every two edges xy,x′y′∈E(P) it satisfies w(xy)≠w(x′y′). The function f is called a rainbow antimagic labeling of G if there exists a rainbow x−y path for every two vertices x,y∈V(G). We say that graph G admits a rainbow antimagic coloring when we assign each edge xy with the color of the edge weight w(xy). The smallest number of colors induced from all edge weights of antimagic labeling is the rainbow antimagic connection number of G, denoted by rac(G). This paper is intended to investigate non-symmetrical phenomena in the comb product of graphs by considering antimagic labeling and optimizing rainbow connection, called rainbow antimagic coloring. In this paper, we show the exact value of the rainbow antimagic connection number of the comb product of graph Fn⊳Tm, where Fn is a friendship graph with order 2n+1 and Tm∈{Pm,Sm,Brm,p,Sm,m}, where Pm is the path graph of order m, Sm is the star graph of order m+1, Brm,p is the broom graph of order m+p and Sm,m is the double star graph of order 2m+2.
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Ly Thi Kieu, Diem, and Nguyen Nguyen Phung. "COHEN-MACAULAYNESS OF SOME EDGE-WEIGHTED GRAPHS." Journal of Science Natural Science 67, no. 3 (October 2022): 17–27. http://dx.doi.org/10.18173/2354-1059.2022-0037.
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In this paper, we will study the characterization of Cohen-Macaulayness of some edge-weighted graphs. For cycle and tree edge-weighted graph, we will reprove the characterization of Cohen-Macaulayness of an edge-weighted cycle and an edge-weighted tree due to C.Paulsen and Wagstaff (2013) [1]. Our proof used a criterion of Hochster for Cohen-Macaulayness of a monomial ideal [2].
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Rasool, Kavi B., Payman A. Rashed, and Ahmed M. Ali. "Relations Between Vertex–Edge Degree Based Topological Indices and Mve-Polynomial of r−Regular Simple Graph." European Journal of Pure and Applied Mathematics 16, no. 2 (April 30, 2023): 773–83. http://dx.doi.org/10.29020/nybg.ejpam.v16i2.4698.
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One of the more exciting polynomials among the newly presented graph algebraic polynomials is the M−Polynomial, which is a standard method for calculating degree−based topological indices. In this paper, we define the Mve−polynomials based on vertex edge degree and derive various vertex–edge degree based topological indices from them. Thus, for any graph, we provide some relationships between vertex–edge degree topological indices. Also, we discuss the general Mve−polynomial of r−regular simple graph. Finally, we computed the Mve−polynomial of the 2−ary tree graph.
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Chen, Bang Ze, and Xiao Bo Yang. "Minimum Spanning Tree Dynamic Demonstration System Implementation." Applied Mechanics and Materials 397-400 (September 2013): 2526–30. http://dx.doi.org/10.4028/www.scientific.net/amm.397-400.2526.
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The graph vertices design into classes, for each vertex in the design of the abscissa, ordinate and in-degree members, realizes the dynamic demonstration minimum spanning tree. Dynamic visualize Prime algorithm and kruskal algorithm implementation process. Around two window synchronization of animation, " in order to find the minimum edge " list box list the minimum edge of a minimum spanning tree ,with thick line in the left window drawing the found minimum edge and On the edge of the vertex, in the right box demo the process of algorithm dynamic execution.
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YANHAONA, MUHAMMAD NUR, MD SHAMSUZZOHA BAYZID, and MD SAIDUR RAHMAN. "DISCOVERING PAIRWISE COMPATIBILITY GRAPHS." Discrete Mathematics, Algorithms and Applications 02, no. 04 (December 2010): 607–23. http://dx.doi.org/10.1142/s1793830910000917.
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Let T be an edge weighted tree, let dT(u, v) be the sum of the weights of the edges on the path from u to v in T, and let d min and d max be two non-negative real numbers such that d min ≤ d max . Then a pairwise compatibility graph of T for d min and d max is a graph G = (V, E), where each vertex u' ∈ V corresponds to a leaf u of T and there is an edge (u', v') ∈ E if and only if d min ≤ dT(u, v) ≤ d max . A graph G is called a pairwise compatibility graph (PCG) if there exists an edge weighted tree T and two non-negative real numbers d min and d max such that G is a pairwise compatibility graph of T for d min and d max . Kearney et al. conjectured that every graph is a PCG [3]. In this paper, we refute the conjecture by showing that not all graphs are PCG s . Moreover, we recognize several classes of graphs as pairwise compatibility graphs. We identify two restricted classes of bipartite graphs as PCG. We also show that the well known tree power graphs and some of their extensions are PCGs.
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Burdonov, Igor Borisovich. "Graph Self-Transformation Model Based on the Operation of Change the End of the Edge." Russian Digital Libraries Journal 23, no. 3 (May 9, 2020): 315–35. http://dx.doi.org/10.26907/1562-5419-2020-23-3-315-335.
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We consider a distributed network whose topology is described by an undirected graph. The network itself can change its topology, using special “commands” provided by its nodes. The work proposes an extremely local atomic transformation acb of a change the end c of the edge ac, “moving” along the edge cb from vertex c to vertex b. As a result of this operation, the edge ac is removed, and the edge ab is added. Such a transformation is performed by a “command” from a common vertex c of two adjacent edges ac and cb. It is shown that from any tree you can get any other tree with the same set of vertices using only atomic transformations. If the degrees of the tree vertices are bounded by the number d (d 3), then the transformation does not violate this restriction. As an example of the purpose of such a transformation, the problems of maximizing and minimizing the Wiener index of a tree with a limited degree of vertices without changing the set of its vertices are considered. The Wiener index is the sum of pairwise distances between the vertices of a graph. The maximum Wiener index has a linear tree (a tree with two leaf vertices). For a root tree with a minimum Wiener index, its type and method for calculating the number of vertices in the branches of the neighbors of the root are determined. Two distributed algorithms are proposed: transforming a tree into a linear tree and transforming a linear tree into a tree with a minimum Wiener index. It is proved that both algorithms have complexity no higher than 2n–2, where n is the number of tree vertices. We also consider the transformation of arbitrary undirected graphs, in which there can be cycles, multiple edges and loops, without restricting the degree of the vertices. It is shown that any connected graph with n vertices can be transformed into any other connected graph with k vertices and the same number of edges in no more than 2(n+k)–2.
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Zhang, Jinshan, Zhengyang Liu, Xiaotie Deng, and Jianwei Yin. "Truthful Mechanisms for Steiner Tree Problems." Proceedings of the AAAI Conference on Artificial Intelligence 37, no. 5 (June 26, 2023): 5884–91. http://dx.doi.org/10.1609/aaai.v37i5.25729.
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Consider an undirected graph G=(V,E) model for a communication network, where each edge is owned by a selfish agent, who reports the cost for offering the use of her edge. Note that each edge agent may misreport her own cost for the use of the edge for her own benefit. In such a non-cooperative setting, we aim at designing an approximately truthful mechanism for establishing a Steiner tree, a minimum cost tree spanning over all the terminals. We present a truthful-in-expectation mechanism that achieves the approximation ratio ln 4 + ε ≈ 1.39, which matches the current best algorithmic ratio for STP.
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Yao, W., P. Polewski, and P. Krzystek. "SEMANTIC LABELLING OF ULTRA DENSE MLS POINT CLOUDS IN URBAN ROAD CORRIDORS BASED ON FUSING CRF WITH SHAPE PRIORS." ISPRS - International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences XLII-2/W7 (September 13, 2017): 971–76. http://dx.doi.org/10.5194/isprs-archives-xlii-2-w7-971-2017.
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In this paper, a labelling method for the semantic analysis of ultra-high point density MLS data (up to 4000 points/m<sup>2</sup>) in urban road corridors is developed based on combining a conditional random field (CRF) for the context-based classification of 3D point clouds with shape priors. The CRF uses a Random Forest (RF) for generating the unary potentials of nodes and a variant of the contrastsensitive Potts model for the pair-wise potentials of node edges. The foundations of the classification are various geometric features derived by means of co-variance matrices and local accumulation map of spatial coordinates based on local neighbourhoods. Meanwhile, in order to cope with the ultra-high point density, a plane-based region growing method combined with a rule-based classifier is applied to first fix semantic labels for man-made objects. Once such kind of points that usually account for majority of entire data amount are pre-labeled; the CRF classifier can be solved by optimizing the discriminative probability for nodes within a subgraph structure excluded from pre-labeled nodes. The process can be viewed as an evidence fusion step inferring a degree of belief for point labelling from different sources. The MLS data used for this study were acquired by vehicle-borne Z+F phase-based laser scanner measurement, which permits the generation of a point cloud with an ultra-high sampling rate and accuracy. The test sites are parts of Munich City which is assumed to consist of seven object classes including impervious surfaces, tree, building roof/facade, low vegetation, vehicle and pole. The competitive classification performance can be explained by the diverse factors: e.g. the above ground height highlights the vertical dimension of houses, trees even cars, but also attributed to decision-level fusion of graph-based contextual classification approach with shape priors. The use of context-based classification methods mainly contributed to smoothing of labelling by removing outliers and the improvement in underrepresented object classes. In addition, the routine operation of a context-based classification for such high density MLS data becomes much more efficient being comparable to non-contextual classification schemes.
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Aji Sailendra, Alfi Istijap, Evawati Alisah, and Achmad Nasichuddin. "DEKOMPOSISI GRAF POHON PISANG Bm,n." Jurnal Riset Mahasiswa Matematika 2, no. 1 (November 1, 2022): 25–31. http://dx.doi.org/10.18860/jrmm.v2i1.14671.
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A decomposition of graph G is collection of subgraphs 〖{H_i}〗_(i=1)^n from G such that H_i [E_i] for E_i is a subset of E(G) and 〖{E_i}〗_(i=1)^n is a partition of E(G). The purpose of the research was to determine the decomposition of the banana tree graph B_(m,n), for m≥1 and n≥2. The research method used in this research is library research. The steps used to determine the decomposition of the banana tree graph B_(m,n) are as follow: (a) Draw a banana tree graph B_(m,n) and label each edge and vertex, (b) Determine the partition on the edges of the banana tree graph B_(m,n), (c) Induced subgraph of from partitions of the banana tree graph B_(m,n), (d) Determine the decomposition of the banana tree graph B_(m,n), (e) Tabulate a conjecture on the decomposition of the banana tree graph B_(m,n), (f) Construct theorem of the decomposition theorem of of the banana tree graph B_(m,n) and its proof. The result of the reasearch is with m≥1 and n≥2, because banana tree graph B_(m,n) is decomposed by the complete graph 〖mK〗_2-decomposition.
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Aji Sailendra, Alfi Istijap, Evawati Alisah, and Achmad Nasichuddin. "DEKOMPOSISI GRAF POHON PISANG Bm,n." Jurnal Riset Mahasiswa Matematika 2, no. 1 (November 1, 2022): 322–28. http://dx.doi.org/10.18860/jrmm.v1i7.14671.
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A decomposition of graph G is collection of subgraphs 〖{H_i}〗_(i=1)^n from G such that H_i [E_i] for E_i is a subset of E(G) and 〖{E_i}〗_(i=1)^n is a partition of E(G). The purpose of the research was to determine the decomposition of the banana tree graph B_(m,n), for m≥1 and n≥2. The research method used in this research is library research. The steps used to determine the decomposition of the banana tree graph B_(m,n) are as follow: (a) Draw a banana tree graph B_(m,n) and label each edge and vertex, (b) Determine the partition on the edges of the banana tree graph B_(m,n), (c) Induced subgraph of from partitions of the banana tree graph B_(m,n), (d) Determine the decomposition of the banana tree graph B_(m,n), (e) Tabulate a conjecture on the decomposition of the banana tree graph B_(m,n), (f) Construct theorem of the decomposition theorem of of the banana tree graph B_(m,n) and its proof. The result of the reasearch is with m≥1 and n≥2, because banana tree graph B_(m,n) is decomposed by the complete graph 〖mK〗_2-decomposition.
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Li, Shu, and Jianfeng Wang. "Yet More Elementary Proof of Matrix-Tree Theorem for Signed Graphs." Algebra Colloquium 30, no. 03 (August 29, 2023): 493–502. http://dx.doi.org/10.1142/s1005386723000408.
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A signed graph [Formula: see text] is a graph [Formula: see text] with vertex set [Formula: see text] and edge set [Formula: see text], together with a function [Formula: see text] assigning a positive or negative sign to each edge. In this paper, we present a more elementary proof for the matrix-tree theorem of signed graphs, which is based on the relations between the incidence matrices and the Laplcians of signed graphs. As an application, we also obtain the results of Monfared and Mallik about the matrix-tree theorem of graphs for signless Laplacians.
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Murugan, A. Nellai, and Shiny Priyanka. "TREE RELATED EXTENDED MEAN CORDIAL GRAPHS." International Journal of Research -GRANTHAALAYAH 3, no. 9 (September 30, 2015): 143–48. http://dx.doi.org/10.29121/granthaalayah.v3.i9.2015.2954.
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Let G = (V,E) be a graph with p vertices and q edges. A Extended Mean Cordial Labeling of a Graph G with vertex set V is a bijection from V to {0, 1,2} such that each edge uv is assigned the label where ⌈ x ⌉ is the least integer greater than or equal to x with the condition that the number of vertices labeled with 0 and the number of vertices labeled with 1 differ by at most 1 and the number of edges labeled with 0 and the number of edges labeled with 1 differ by almost 1. The graph that admits an Extended Mean Cordial Labeling is called Extended Mean Cordial Graph. In this paper, we proved that tree related graphs Hdn, K 1,n, Tgn, <K1,n:n> are Extended Mean Cordial Graphs.
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Qiu, Teng, and Yongjie Li. "Nearest Descent, In-Tree, and Clustering." Mathematics 10, no. 5 (February 27, 2022): 764. http://dx.doi.org/10.3390/math10050764.
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Clustering aims at discovering the natural groupings in a dataset, prevalent in many disciplines that involve multivariate data analysis. In this paper, we propose a physically inspired graph-theoretical clustering method, which first makes the data points organized into an attractive graph, called In-Tree, via a physically inspired rule, called Nearest Descent (ND). The rule of ND works to select the nearest node in the descending direction of potential as the parent node of each node, which is fundamentally different from the classical Gradient Descent. The constructed In-Tree proves a very good candidate for clustering due to its particular features and properties. In the In-Tree, the original clustering problem is reduced to a problem of removing the inter-cluster edges from this graph. Pleasingly, those inter-cluster edges are usually so distinguishable that they can be easily determined by different automatic edge-cutting methods. We also propose a visualized strategy to validate the effectiveness of the automatic edge-cutting methods. The experimental results reveal that the proposed method is superior to the related clustering methods. The results also reveal the characteristics of different automatic cutting methods and the meaningfulness of the visualized strategy in increasing the reliability of the clustering results in practice.
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Kristiana, Arika Indah, Ahmad Aji, Edy Wihardjo, and Deddy Setiawan. "on Graceful Chromatic Number of Vertex amalgamation of Tree Graph Family." CAUCHY: Jurnal Matematika Murni dan Aplikasi 7, no. 3 (October 11, 2022): 432–44. http://dx.doi.org/10.18860/ca.v7i3.16334.
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Proper vertex coloring c of a graph G is a graceful coloring if c is a graceful k-coloring for k∈{1,2,3,…}. Definition graceful k-coloring of a graph G=(V,E) is a proper vertex coloring c:V(G)→{1,2,…,k);k≥2, which induces a proper edge coloring c':E(G)→{1,2,…,k-1} defined c'(uv)=|c(u)-c(v)|. The minimum vertex coloring from graph G can be colored with graceful coloring called a graceful chromatic number with notation χg (G). In this paper, we will investigate the graceful chromatic number of vertex amalgamation of tree graph family with some graph is path graph, centipede graph, broom and E graph.
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SUN, YUEFANG. "The (3, l)-Rainbow Edge-Index of Cartesian Product Graphs." Journal of Interconnection Networks 17, no. 03n04 (September 2017): 1741009. http://dx.doi.org/10.1142/s0219265917410092.
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For a graph G and a vertex subset [Formula: see text] of at least two vertices, an S-tree is a subgraph T of G that is a tree with [Formula: see text]. Two S-trees are said to be edge-disjoint if they have no common edge. Let [Formula: see text] denote the maximum number of edge-disjoint S-trees in G. For an integer K with [Formula: see text], the generalized k-edge-connectivity is defined as [Formula: see text]. An S-tree in an edge-colored graph is rainbow if no two edges of it are assigned the same color. Let [Formula: see text] and l be integers with [Formula: see text], the [Formula: see text]-rainbow edge-index [Formula: see text] of G is the smallest number of colors needed in an edge-coloring of G such that for every set S of k vertices of G, there exist l edge-disjoint rainbow S-trees.In this paper, we study the [Formula: see text]-rainbow edge-index of Cartesian product graphs and get a sharp upper bound for [Formula: see text] , where G and H are connected graphs with orders at least three, and [Formula: see text] denotes the Cartesian product of G and H.
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Zhou, Jian, Lu Chen, and Ke Wang. "Path Optimality Conditions for Minimum Spanning Tree Problem with Uncertain Edge Weights." International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems 23, no. 01 (February 2015): 49–71. http://dx.doi.org/10.1142/s0218488515500038.
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This paper investigates the uncertain minimum spanning tree (UMST) problem where the edge weights are assumed to be uncertain variables. In order to propose effective solving methods for the UMST problem, path optimality conditions as well as some equivalent definitions for two commonly used types of UMST, namely, uncertain expected minimum spanning tree (expected UMST) and uncertain α-minimum spanning tree (α-UMST), are discussed. It is shown that both the expected UMST problem and the α-UMST problem can be transformed into an equivalent classical minimum spanning tree problem on a corresponding deterministic graph, which leads to effective algorithms with low computational complexity. Furthermore, the notion of uncertain most minimum spanning tree (most UMST) is initiated for an uncertain graph, and then the equivalent relationship between the α-UMST and the most UMST is proved. Numerical examples are presented as well for illustration.
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Ghanbari, Nima, and Saeid Alikhani. "A graph related to the Euler ø function." Mathematical Gazette 107, no. 569 (July 2023): 263–72. http://dx.doi.org/10.1017/mag.2023.57.
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In mathematics, graph theory is the study of graphs, which are mathematical structures used to model pairwise relations between objects. A graph G is a pair G = (V, E), where V and E are the vertex set and the edge set of G, respectively. The order and size of G is the number of vertices and edges of G, respectively. The degree or valency of a vertex u in a graph G (loopless), denoted by deg (u), is the number of edges meeting at u. If, for every vertex ν in G, deg (ν) = k, we say that G is a k-regular graph. The cycle of order n is denoted by Cn and is a connected 2-regular graph. The path graph of order n is denoted by Pn and obtain by deleting an edge of Cn. A tree is an undirected graph in which any two vertices are connected by exactly one path, or equivalently a connected undirected graph without cycle. A leaf (or pendant vertex) of a tree is a vertex of the tree of degree 1. An edge of a graph is said to be pendant if one of its vertices is a pendant vertex. A complete bipartite graph is a graph G with and such that every vertex of the set (part) X is connected to every vertex of the set (part) Y. If , then this graph is denoted by Km,n. The complete bipartite graph K1,n is called the star graph which has n + 1 vertices. The distance between two vertices u and ν of G, denoted by d (u, ν), is defined as the minimum number of edges of the walks between them. The complement of graph G is denoted by and is a graph with the same vertices such that two distinct vertices of are adjacent if, and only if, they are not adjacent in G. For more information on graphs, refer to [1].
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Li, Pingshan, Rong Liu, and Xianglin Liu. "The (E)FTSM-(edge) Connectivity of Cayley Graphs Generated by Transposition Trees." International Journal of Foundations of Computer Science 33, no. 01 (October 14, 2021): 33–43. http://dx.doi.org/10.1142/s0129054121500349.
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The Cayley graph generated by a transposition tree [Formula: see text] is a class of Cayley graphs that contains the star graph and the bubble sort graph. A graph [Formula: see text] is called strongly Menger (SM for short) (edge) connected if each pair of vertices [Formula: see text] are connected by [Formula: see text] (edge)-disjoint paths, where [Formula: see text] are the degree of [Formula: see text] and [Formula: see text] respectively. In this paper, the maximally edge-fault-tolerant and the maximally vertex-fault-tolerant of [Formula: see text] with respect to the SM-property are found and thus generalize or improve the results in [19, 20, 22, 26] on this topic.
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Li, Pingshan, Rong Liu, and Xianglin Liu. "The (E)FTSM-(edge) Connectivity of Cayley Graphs Generated by Transposition Trees." International Journal of Foundations of Computer Science 33, no. 01 (October 14, 2021): 33–43. http://dx.doi.org/10.1142/s0129054121500349.
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The Cayley graph generated by a transposition tree [Formula: see text] is a class of Cayley graphs that contains the star graph and the bubble sort graph. A graph [Formula: see text] is called strongly Menger (SM for short) (edge) connected if each pair of vertices [Formula: see text] are connected by [Formula: see text] (edge)-disjoint paths, where [Formula: see text] are the degree of [Formula: see text] and [Formula: see text] respectively. In this paper, the maximally edge-fault-tolerant and the maximally vertex-fault-tolerant of [Formula: see text] with respect to the SM-property are found and thus generalize or improve the results in [19, 20, 22, 26] on this topic.
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Ferrarello, Daniela. "The complement of a d-tree is Cohen-Macaulay." MATHEMATICA SCANDINAVICA 99, no. 2 (December 1, 2006): 161. http://dx.doi.org/10.7146/math.scand.a-15006.
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In this work we obtain the result that the complement of a $d$-tree is a Cohen-Macaulay graph. To do this we use a theorem by Fröberg that estabilishes a condition for a Stanley-Reisner ring of a simplicial complex to be Cohen-Macaulay and a useful lemma to pass from a Stanley-Reisner ideal of a simplicial complex to an edge ideal of a graph.
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Casinillo, Emily L., and Leomarich F. Casinillo. "A Note on Full and Complete Binary Planar Graphs." Indonesian Journal of Mathematics Education 3, no. 2 (October 30, 2020): 70. http://dx.doi.org/10.31002/ijome.v3i2.2800.
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<p>Let G=(V(G), E(G)) be a connected graph where V(G) is a finite nonempty set called vertex-set of G, and E(G) is a set of unordered pairs {u, v} of distinct elements from V(G) called the edge-set of G. If is a connected acyclic graph or a connected graph with no cycles, then it is called a tree graph. A binary tree Tl with l levels is complete if all levels except possibly the last are completely full, and the last level has all its nodes to the left side. If we form a path on each level of a full and complete binary tree, then the graph is now called full and complete binary planar graph and it is denoted as Bn, where n is the level of the graph. This paper introduced a new planar graph which is derived from binary tree graphs. In addition, a combinatorial formula for counting its vertices, faces, and edges that depends on the level of the graph was developed.</p>
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Rahman, Md Zamilur, Asish Mukhopadhyay, and Yash P. Aneja. "A separator-based method for generating weakly chordal graphs." Discrete Mathematics, Algorithms and Applications 12, no. 04 (July 23, 2020): 2050039. http://dx.doi.org/10.1142/s1793830920500391.
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We propose a scheme for generating a weakly chordal graph on [Formula: see text] vertices with [Formula: see text] edges. In this method, we first construct a tree and then generate an orthogonal layout (which is a weakly chordal graph on the [Formula: see text] vertices) based on this tree. We then insert additional edges, if needed, for a total of [Formula: see text] edges. Our algorithm ensures that the graph remains weakly chordal after each edge is inserted. The time complexity of an insertion query is [Formula: see text], where [Formula: see text] and [Formula: see text] are the degrees of the vertices [Formula: see text] and [Formula: see text] we want to join with an edge and an insertion takes constant time. The advantages of this method are that it uses very simple data structures and exploits the basic structural properties of a weakly chordal graph.
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El-Zohny, H., S. Radwan, S. I. Abo El-Fotooh, and Z. Mohammed. "Graceful Labeling of Hypertrees." Journal of Mathematics Research 13, no. 1 (January 11, 2021): 28. http://dx.doi.org/10.5539/jmr.v13n1p28.
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Graph labeling is considered as one of the most interesting areas in graph theory. A labeling for a simple graph G (numbering or valuation), is an association of non -negative integers to vertices of G (vertex labeling) or to edges of G (edge labeling) or both of them. In this paper we study the graceful labeling for the k- uniform hypertree and define a condition for the corresponding tree to be graceful. A k- uniform hypertree is graceful if the minimum difference of vertices’ labels of each edge is distinct and each one is the label of the corresponding edge.
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El-Zohny, H., S. Radwan, S. I. Abo El-Fotooh, and Z. Mohammed. "Graceful Labeling of Hypertrees." Journal of Mathematics Research 13, no. 1 (January 11, 2021): 28. http://dx.doi.org/10.5539/jmr.v13n1p28.
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Graph labeling is considered as one of the most interesting areas in graph theory. A labeling for a simple graph G (numbering or valuation), is an association of non -negative integers to vertices of G (vertex labeling) or to edges of G (edge labeling) or both of them. In this paper we study the graceful labeling for the k- uniform hypertree and define a condition for the corresponding tree to be graceful. A k- uniform hypertree is graceful if the minimum difference of vertices’ labels of each edge is distinct and each one is the label of the corresponding edge.
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Montgomery, R., A. Pokrovskiy, and B. Sudakov. "A proof of Ringel’s conjecture." Geometric and Functional Analysis 31, no. 3 (June 2021): 663–720. http://dx.doi.org/10.1007/s00039-021-00576-2.
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AbstractA typical decomposition question asks whether the edges of some graph G can be partitioned into disjoint copies of another graph H. One of the oldest and best known conjectures in this area, posed by Ringel in 1963, concerns the decomposition of complete graphs into edge-disjoint copies of a tree. It says that any tree with n edges packs $$2n+1$$ 2 n + 1 times into the complete graph $$K_{2n+1}$$ K 2 n + 1 . In this paper, we prove this conjecture for large n.
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Gochev, Kalin, Alla Safonova, and Maxim Likhachev. "Anytime Tree-Restoring Weighted A* Graph Search." Proceedings of the International Symposium on Combinatorial Search 5, no. 1 (September 1, 2021): 80–88. http://dx.doi.org/10.1609/socs.v5i1.18321.
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Incremental graph search methods reuse information from previous searches in order to minimize redundant computation and to find solutions to series of similar search queries much faster than it is possible by solving each query from scratch. In this work, we present a simple, but very effective, technique for performing incremental weighted A* graph search in an anytime fashion. On the theoretical side, we show that our anytime incremental algorithm preserves the strong theoretical guarantees provided by the weighted A* algorithm, such as completeness and bounds on solution cost sub-optimality. We also show that our algorithm can handle a variety of changes to the underlying graph, such as both increasing and decreasing edge costs, and changes in the heuristic. On the experimental side, we demonstrate the effectiveness of our algorithm in the context of (x, y, z, yaw) navigation planning for an unmanned aerial vehicle and compare our algorithm to popular incremental and anytime graph search algorithms.
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Gabrysch, Katja. "Convergence of directed random graphs to the Poisson-weighted infinite tree." Journal of Applied Probability 53, no. 2 (June 2016): 463–74. http://dx.doi.org/10.1017/jpr.2016.13.
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Abstract We consider a directed graph on the integers with a directed edge from vertex i to j present with probability n-1, whenever i < j, independently of all other edges. Moreover, to each edge (i, j) we assign weight n-1(j - i). We show that the closure of vertex 0 in such a weighted random graph converges in distribution to the Poisson-weighted infinite tree as n → ∞. In addition, we derive limit theorems for the length of the longest path in the subgraph of the Poisson-weighted infinite tree which has all vertices at weighted distance of at most ρ from the root.
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Kwun, Young Chel, Hafiz Mutee ur Rehman, Muhammad Yousaf, Waqas Nazeer, and Shin Min Kang. "The Entropy of Weighted Graphs with Atomic Bond Connectivity Edge Weights." Discrete Dynamics in Nature and Society 2018 (December 16, 2018): 1–10. http://dx.doi.org/10.1155/2018/8407032.
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The aim of this report to solve the open problem suggested by Chen et al. We study the graph entropy with ABC edge weights and present bounds of it for connected graphs, regular graphs, complete bipartite graphs, chemical graphs, tree, unicyclic graphs, and star graphs. Moreover, we compute the graph entropy for some families of dendrimers.
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Marsidi, Ika Hesti Agustin, Dafik, Elsa Yuli Kurniawati, and Rosanita Nisviasari. "The rainbow vertex antimagic coloring of tree graphs." Journal of Physics: Conference Series 2157, no. 1 (January 1, 2022): 012019. http://dx.doi.org/10.1088/1742-6596/2157/1/012019.
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Abstract Let G(V (G),E(G)) be a connected, simple, and finite graph. Let f be a bijective function of labeling on graph G from the edge set E(G) to natural number up to the number of edges of G. A rainbow vertex antimagic labeling of graph G is a function f under the condition all internal vertices of a path u – υ, Ɐu, υ ∈ V (G) have different weight (denoted by w(u)), where w(u) = ∑ uu′∈E(G)f (uu′). If G has a rainbow vertex antimagic labeling, then G is a rainbow vertex antimagic coloring, where the every vertex is assigned with the color w(u). The rvac(G) is a notation of rainbow vertex antimagic connection number of graph G which means the minimum colors taken over all rainbow vertex antimagic coloring induced by rainbow vertex antimagic labeling of graph G. The results of this research are the exact value of the rainbow vertex antimagic connection number of star (Sn ), double star (DSn ), and broom graph (Brn, m ).