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Статті в журналах з теми "Graph"

Автор: Grafiati
Опубліковано: 4 червня 2021
Оновлено: 9 червня 2025

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1

CSIKVÁRI, PÉTER, and ZOLTÁN LÓRÁNT NAGY. "The Density Turán Problem." Combinatorics, Probability and Computing 21, no. 4 (February 29, 2012): 531–53. http://dx.doi.org/10.1017/s0963548312000016.

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Анотація:
LetHbe a graph onnvertices and let the blow-up graphG[H] be defined as follows. We replace each vertexviofHby a clusterAiand connect some pairs of vertices ofAiandAjif (vi,vj) is an edge of the graphH. As usual, we define the edge density betweenAiandAjasWe study the following problem. Given densities γijfor each edge (i,j) ∈E(H), one has to decide whether there exists a blow-up graphG[H], with edge densities at least γij, such that one cannot choose a vertex from each cluster, so that the obtained graph is isomorphic toH,i.e., noHappears as a transversal inG[H]. We calldcrit(H) the maximal value for which there exists a blow-up graphG[H] with edge densitiesd(Ai,Aj)=dcrit(H) ((vi,vj) ∈E(H)) not containingHin the above sense. Our main goal is to determine the critical edge density and to characterize the extremal graphs.First, in the case of treeTwe give an efficient algorithm to decide whether a given set of edge densities ensures the existence of a transversalTin the blow-up graph. Then we give general bounds ondcrit(H) in terms of the maximal degree. In connection with the extremal structure, the so-called star decomposition is proved to give the best construction forH-transversal-free blow-up graphs for several graph classes. Our approach applies algebraic graph-theoretical, combinatorial and probabilistic tools.
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2

Cappelletti, Luca, Tommaso Fontana, Elena Casiraghi, Vida Ravanmehr, Tiffany J. Callahan, Carlos Cano, Marcin P. Joachimiak, et al. "GRAPE for fast and scalable graph processing and random-walk-based embedding." Nature Computational Science 3, no. 6 (June 26, 2023): 552–68. http://dx.doi.org/10.1038/s43588-023-00465-8.

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Анотація:
AbstractGraph representation learning methods opened new avenues for addressing complex, real-world problems represented by graphs. However, many graphs used in these applications comprise millions of nodes and billions of edges and are beyond the capabilities of current methods and software implementations. We present GRAPE (Graph Representation Learning, Prediction and Evaluation), a software resource for graph processing and embedding that is able to scale with big graphs by using specialized and smart data structures, algorithms, and a fast parallel implementation of random-walk-based methods. Compared with state-of-the-art software resources, GRAPE shows an improvement of orders of magnitude in empirical space and time complexity, as well as competitive edge- and node-label prediction performance. GRAPE comprises approximately 1.7 million well-documented lines of Python and Rust code and provides 69 node-embedding methods, 25 inference models, a collection of efficient graph-processing utilities, and over 80,000 graphs from the literature and other sources. Standardized interfaces allow a seamless integration of third-party libraries, while ready-to-use and modular pipelines permit an easy-to-use evaluation of graph-representation-learning methods, therefore also positioning GRAPE as a software resource that performs a fair comparison between methods and libraries for graph processing and embedding.
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3

Liu, Yu, and Lihua You. "Further Results on the Nullity of Signed Graphs." Journal of Applied Mathematics 2014 (2014): 1–8. http://dx.doi.org/10.1155/2014/483735.

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Анотація:
The nullity of a graph is the multiplicity of the eigenvalue zero in its spectrum. A signed graph is a graph with a sign attached to each of its edges. In this paper, we apply the coefficient theorem on the characteristic polynomial of a signed graph and give two formulae on the nullity of signed graphs with cut-points. As applications of the above results, we investigate the nullity of the bicyclic signed graphΓ∞p,q,l, obtain the nullity set of unbalanced bicyclic signed graphs, and thus determine the nullity set of bicyclic signed graphs.
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4

Ji, Shengwei, Chenyang Bu, Lei Li, and Xindong Wu. "Local Graph Edge Partitioning." ACM Transactions on Intelligent Systems and Technology 12, no. 5 (October 31, 2021): 1–25. http://dx.doi.org/10.1145/3466685.

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Анотація:
Graph edge partitioning, which is essential for the efficiency of distributed graph computation systems, divides a graph into several balanced partitions within a given size to minimize the number of vertices to be cut. Existing graph partitioning models can be classified into two categories: offline and streaming graph partitioning models. The former requires global graph information during the partitioning, which is expensive in terms of time and memory for large-scale graphs. The latter creates partitions based solely on the received graph information. However, the streaming model may result in a lower partitioning quality compared with the offline model. Therefore, this study introduces a Local Graph Edge Partitioning model, which considers only the local information (i.e., a portion of a graph instead of the entire graph) during the partitioning. Considering only the local graph information is meaningful because acquiring complete information for large-scale graphs is expensive. Based on the Local Graph Edge Partitioning model, two local graph edge partitioning algorithms—Two-stage Local Partitioning and Adaptive Local Partitioning—are given. Experimental results obtained on 14 real-world graphs demonstrate that the proposed algorithms outperform rival algorithms in most tested cases. Furthermore, the proposed algorithms are proven to significantly improve the efficiency of the real graph computation system GraphX.
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5

Kok, Johan. "Note: Certain bounds in respect of upper deg-centric graphs." Open Journal of Discrete Applied Mathematics 7, no. 2 (December 31, 2024): 1–6. https://doi.org/10.30538/psrp-odam2024.0097.

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Анотація:
This note presents some upper bounds for the size of the upper deg-centric grapg \(G_{ud}\) of a simple connected graph G. Amongst others, a result for graphs for which a compliant graph \(G\) has \(G_{ud} \cong \overline G\) is presented. Finally, results for size minimality in respect upper deg-centrication and minimum size of such graph \(G\) are presented.
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6

Duan, Yucong, Lixu Shao, and Gongzhu Hu. "Specifying Knowledge Graph with Data Graph, Information Graph, Knowledge Graph, and Wisdom Graph." International Journal of Software Innovation 6, no. 2 (April 2018): 10–25. http://dx.doi.org/10.4018/ijsi.2018040102.

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Анотація:
Knowledge graphs have been widely adopted, in large part owing to their schema-less nature. It enables knowledge graphs to grow seamlessly and allows for new relationships and entities as needed. A knowledge graph is a graph constructed by representing each item, entity and user as nodes, and linking those nodes that interact with each other via edges. Knowledge graphs have abundant natural semantics and can contain various and more complete information. It is an expression mechanism close to natural language. However, we still lack a unified definition and standard expression form of knowledge graph. The authors propose to clarify the expression of knowledge graph as a whole. They clarify the architecture of knowledge graph from data, information, knowledge, and wisdom aspects respectively. The authors also propose to specify knowledge graph in a progressive manner as four basic forms including data graph, information graph, knowledge graph and wisdom graph.
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7

Sohn, Moo Young, and Jaeun Lee. "Characteristic polynomials of some weighted graph bundles and its application to links." International Journal of Mathematics and Mathematical Sciences 17, no. 3 (1994): 503–10. http://dx.doi.org/10.1155/s0161171294000748.

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Анотація:
In this paper, we introduce weighted graph bundles and study their characteristic polynomial. In particular, we show that the characteristic polynomial of a weightedK2(K¯2)-bundles over a weighted graphG?can be expressed as a product of characteristic polynomials two weighted graphs whose underlying graphs areGAs an application, we compute the signature of a link whose corresponding weighted graph is a double covering of that of a given link.
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8

JOHANNSEN, DANIEL, MICHAEL KRIVELEVICH, and WOJCIECH SAMOTIJ. "Expanders Are Universal for the Class of All Spanning Trees." Combinatorics, Probability and Computing 22, no. 2 (January 3, 2013): 253–81. http://dx.doi.org/10.1017/s0963548312000533.

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Анотація:
A graph is calleduniversalfor a given graph class(or, equivalently,-universal) if it contains a copy of every graph inas a subgraph. The construction of sparse universal graphs for various classeshas received a considerable amount of attention. There is particular interest in tight-universal graphs, that is, graphs whose number of vertices is equal to the largest number of vertices in a graph from. Arguably, the most studied case is that whenis some class of trees. In this work, we are interested in(n,Δ), the class of alln-vertex trees with maximum degree at most Δ. We show that everyn-vertex graph satisfying certain natural expansion properties is(n,Δ)-universal. Our methods also apply to the case when Δ is some function ofn. Since random graphs are known to be good expanders, our result implies, in particular, that there exists a positive constantcsuch that the random graphG(n,cn−1/3log2n) is asymptotically almost surely (a.a.s.) universal for(n,O(1)). Moreover, a corresponding result holds for the random regular graph of degreecn2/3log2n. Another interesting consequence is the existence of locally sparsen-vertex(n,Δ)-universal graphs. For example, we show that one can (randomly) constructn-vertex(n,O(1))-universal graphs with clique number at most five. This complements the construction of Bhatt, Chung, Leighton and Rosenberg (1989), whose(n,Δ)-universal graphs with merelyO(n)edges contain large cliques of size Ω(Δ). Finally, we show that random graphs are robustly(n,Δ)-universal in the context of the Maker–Breaker tree-universality game.
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9

Kaviya, S., G. Mahadevan, and C. Sivagnanam. "Generalizing TCCD-Number For Power Graph Of Some Graphs." Indian Journal Of Science And Technology 17, SPI1 (April 25, 2024): 115–23. http://dx.doi.org/10.17485/ijst/v17sp1.243.

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Анотація:
Objective: Finding the triple connected certified domination number for the power graph of some peculiar graphs. Methods: A dominating set with the condition that every vertex in has either zero or at least two neighbors in and is triple connected is a called triple connected certified domination number of a graph. The minimum cardinality among all the triple connected certified dominating sets is called the triple connected certified domination number and is denoted by . The upper bound and lower bound of for the given graphs is found and then proved the upper bound and lower bound of were equal. Findings: We found the (TCCD)-number for the power graph of some peculiar graphs. Also, we have generalized the result for path, cycle, ladder graph, comb graph, coconut tree graph, triangular snake, alternate triangular snake, quadrilateral snake and tadpole graph. Novelty: The triple connected certified domination is a new parameter in which the certified domination holds the triple connected in induced . Keywords: Domination Number, Power Graphs, Triple Connected, Certified Domination, Triple Connected Certified Domination
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10

Simonet, Geneviève, and Anne Berry. "Properties and Recognition of Atom Graphs." Algorithms 15, no. 8 (August 19, 2022): 294. http://dx.doi.org/10.3390/a15080294.

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Анотація:
The atom graph of a connected graph is a graph whose vertices are the atoms obtained by clique minimal separator decomposition of this graph, and whose edges are the edges of all its atom trees. A graph G is an atom graph if there is a graph whose atom graph is isomorphic to G. We study the class of atom graphs, which is also the class of atom graphs of chordal graphs, and the associated recognition problem. We prove that each atom graph is a perfect graph and give a characterization of atom graphs in terms of a spanning tree, inspired by the characterization of clique graphs of chordal graphs as expanded trees. We also characterize the chordal graphs having the same atom and clique graph, and solve the recognition problem of atom graphs of two graph classes.
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11

Wafiq, Hibi. "Non-Isomorphism Between Graph And Its Complement." Multicultural Education 7, no. 6 (June 16, 2021): 256. https://doi.org/10.5281/zenodo.4965942.

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Анотація:
<em>It is known that any graph with six vertices cannot be isomorphic to its complement [3].V. K. Balakrishnan has written in his book Schaum&rsquo;s solved problems series [1] the following: &ldquo;Given two arbitrary Simple graphs of the same order and the same size, the problem of determining whetheran isomorphism exists between the two is known as the isomorphism problem in graph theory. In general, itis not all easy (in other words, there is no &quot;efficient algorithm&quot;) to solve an arbitrary instance of the isomorphismproblem&rdquo;, from here came the idea of this paper. As mentioned,isomorphism between two graphs G_1and G_2 is a one-valued function that copies the vertices of graph G_1over all the vertices of graphG_2. If two vertices connected on edge in graphG_1, their imageswill connected on edge in graphG_2;also, if two vertices are not connected in edge on graph G_1 so their images will also not be connected in edge also on graphG_2.The complementary graph of graph G and which will be marked with G ̅, is defined as the graph that contains the same set of vertices, and two vertices connected on edge in graph G will be connected on edge in graph G ̅, and two non-connected vertices on edge in graph G will be non-connected vertices on edge in graphG ̅. The purpose of this paper is to present anenough condition on the vertex number of a given graph so that it is not isomorphic to its complement.</em>
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12

Basavanagoud, B., and Roopa S. Kusugal. "On the Line Degree Splitting Graph of a Graph." Bulletin of Mathematical Sciences and Applications 18 (May 2017): 1–10. http://dx.doi.org/10.18052/www.scipress.com/bmsa.18.1.

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Анотація:
In this paper, we introduce the concept of the line degree splitting graph of a graph. We obtain some properties of this graph. We find the girth of the line degree splitting graphs. Further, we establish the characterization of graphs whose line degree splitting graphs are eulerian, complete bipartite graphs and complete graphs.
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13

Demirci, Musa, Sadik Delen, Ahmet Sinan Cevik, and Ismail Naci Cangul. "Omega Index of Line and Total Graphs." Journal of Mathematics 2021 (September 27, 2021): 1–6. http://dx.doi.org/10.1155/2021/5552202.

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Анотація:
A derived graph is a graph obtained from a given graph according to some predetermined rules. Two of the most frequently used derived graphs are the line graph and the total graph. Calculating some properties of a derived graph helps to calculate the same properties of the original graph. For this reason, the relations between a graph and its derived graphs are always welcomed. A recently introduced graph index which also acts as a graph invariant called omega is used to obtain such relations for line and total graphs. As an illustrative exercise, omega values and the number of faces of the line and total graphs of some frequently used graph classes are calculated.
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14

Bursal, Murat, and Serap Yetiş. "Middle School Students’ Graph Skills and Affective States about Graphs." International Journal of Research in Education and Science 6, no. 4 (September 19, 2020): 692. http://dx.doi.org/10.46328/ijres.v6i4.1136.

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Анотація:
This survey design study was designed to test whether the graph skills and affective states of middle school students about graphs differ by their gender, grade level, and graph types (line, bar, and pie). The data collection instruments consisted of two scales developed by the authors and a Graph Skills Test, which consisted of graph questions from the previous TIMSS and PISA exams. Based on the findings, while middle school students were found to succeed at reading the data level graph questions, they were found to struggle in questions requiring higher graph skills, such as graph interpretation and graph construction. As for the affective states investigated, participants were found to hold high self-efficacy beliefs and positive attitudes toward graphs. No significant difference among the dependent variables (graph skills, self-efficacy beliefs about graphs, attitudes toward graphs, and graph literacy perceptions) was found by gender; however, grade level and graph type variables were found to impact students’ graph skills, graph attitudes, and personal graph literacy perceptions. Middle school students with less school experience with graphs (seventh graders) were found to hold more positive attitudes toward graphs than the eighth graders. On the contrary, eighth graders were found to perform better at graph questions requiring interpretations of the graph data. Also, participants in all subgroups were found to hold significantly higher personal graph literacy perceptions for the bar graphs, than the line graphs and pie charts. Based on the findings of the study, while middle school students were found to hold positive affective states about graphs, they were found to lack advanced graph skills. In agreement with the previous literature, it is recommended that graph literacy should become a dedicated part of the school curriculum.
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15

Lakshmanan S., Aparna, S. B. Rao, and A. Vijayakumar. "Gallai and anti-Gallai graphs of a graph." Mathematica Bohemica 132, no. 1 (2007): 43–54. http://dx.doi.org/10.21136/mb.2007.133996.

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16

Radha, K., and A. Jasmine Kingsly. "Cardinality and Isomorphic Properties of Hajós Graphs and Hajós Fuzzy Graphs." Indian Journal Of Science And Technology 17, no. 37 (October 5, 2024): 3871–80. http://dx.doi.org/10.17485/ijst/v17i37.2492.

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Анотація:
Objectives: Fuzzy graphs allow uncertainty in the ideas characterizing vertices and edges to be included when modelling real-world scenarios into graph models. This study aims to introduce Hajós fuzzy graph, cardinality of Hajós graph and Hajós fuzzy graph and to explore many properties. Methods : Here, the Hajós construction is applied to two fuzzy graphs, and the Hajós fuzzy graph is defined by assigning membership values to all the edges and vertices of the newly formed fuzzy graph. Fuzzy graph conditions are verified for the assigned membership values. Examples are used to illustrate the concept. Findings: The Hajós fuzzy graph's size and order are established. Isomorphic property is discussed. The number of Hajós (fuzzy) graphs that exist for some combinations of two (fuzzy) graphs is calculated. Novelty: Based on Hajós construction, the Hajós fuzzy graph is defined. Based on the fact that different choices of vertices and edges produce different Hajós (fuzzy) graph, its cardinality is defined. The notion of cardinality is a novel concept in Hajos graph which helps to find the total number of Hajós (fuzzy) graphs. The cardinality of the Hajós (fuzzy) graph which arises from two (fuzzy) graphs that can be a cycle, path, regular graph, complete graph and complete bipartite graph are derived. Mathematics subject classification. (2010): 05C72, 05C76 Keywords: Hajós Graph, Hajós fuzzy graph, Order, Size, Isomorphism, Cardinality of Hajós fuzzy graph
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17

Sufia Aziz. "Characteristic Graph vs Benzenoid Graph." Mathematical Journal of Interdisciplinary Sciences 7, no. 2 (March 6, 2019): 135–47. http://dx.doi.org/10.15415/mjis.2019.72018.

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Анотація:
A characteristic graph is a tree representative of its corresponding benzenoid (cyclic) graph. It may contain necessary information of several properties of benzenoids. The PI-index of benzenoids and their characteristic graphs are compared by correlating it to a structural property (π-electron energy) of the benzenoids using MLR analysis. PI index being applicable to both trees and cyclic graphs, yielded required results for benzenoid and their characteristic graph.
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18

S, Balakrishnan, Mohamed Ali A, and Rama Murthi K. "A Study on Different types of Product of Anti Fuzzy Graphs." Journal of Computational Mathematica 7, no. 1 (June 30, 2023): 131–38. http://dx.doi.org/10.26524/cm168.

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Анотація:
In this article, we consider to obtain new anti fuzzy graph from given anti fuzzy graphs. Product of anti fuzzy graphs is an operation on anti fuzzy graphs that produce new anti fuzzy graph. We consider tensor, normal, modular anti fuzzy graphs products, which are adapted from fuzzy graphs products. In general, product of any two anti fuzzy graphs is an anti fuzzy graph. Product of any two strong anti fuzzy graphs is an anti fuzzy graph strong. Normal product of two anti fuzzy graph complete is a complete anti fuzzy graph. Other than that, different of anti fuzzy products are also discussed.
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19

Et. al., PuruchothamaNayakiM. "Distance Based Topological Indices And Regular Graphs." Turkish Journal of Computer and Mathematics Education (TURCOMAT) 12, no. 3 (April 11, 2021): 5191–96. http://dx.doi.org/10.17762/turcomat.v12i3.2147.

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Анотація:
In this article, we are using the regular graph of even number of vertices and computing the distance balanced graphs. First we take a graph for satisfying regular definition and then we compute the Mostar index of that particular graph. If the Mostar index of that particular graph is zero, then the graph is said to be a distance balanced graph. So we discuss first distance balanced graph. Suppose if we delete one edge in that particular graph, that is non-regular graph, we can verify the balanced graph is whether distance balanced graph or not. We discuss and compute the Mostar index of certain regular and non-regular graphs are balanced distance or not. Finally we see few theorems are related in this topic. So in this paper, we study some distance based topological indices for regular graphs and also cubic graphs.
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20

Zeen El Deen, Mohamed R. "Enumeration of spanning trees in prisms of some graphs." Proyecciones (Antofagasta) 42, no. 2 (December 1, 2023): 339–91. http://dx.doi.org/10.22199/issn.0717-6279-4664.

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Анотація:
In graph theory, a prism over a graph G is the cartesian product of the graph G with P₂. The purpose of this work is to investigate the complexity of the prisms of some path and cycle-related graphs. In particular, we obtain simpler and more explicit formulas for the complexity of a special class of prisms of path-related graphs: fan graph, ladder graph, the composition Pn[P₂] graph, and book graph. Moreover, we obtain straightforward formulas for the complexity of a special class of prisms of cycle-related graphs: wheel graph, gear graph, prism graph, n−crossed prism graph, mirror graph M(Cn) of even cycle Cn, twisted prism, total graph T(Cn) of the cycle Cn, the friendship graph, the flower graph, and planner sunflower graph. These closed formulas are deduced using some basic properties of block matrix, recurrence relation, eigenvalues of circulant matrices, and orthogonal polynomials.
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21

Fujita, Takaaki. "Claw-free Graph and AT-free Graph in Fuzzy, Neutrosophic, and Plithogenic Graphs." Information Sciences with Applications 5 (March 5, 2025): 40–55. https://doi.org/10.61356/j.iswa.2025.5502.

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Анотація:
Graph theory studies networks consisting of nodes (vertices) and their connections (edges), with various graph classes being extensively researched. This paper focuses on three specific graph classes: AT-Free Graphs, Claw-Free Graphs, and Triangle-Free Graphs. Additionally, it examines uncertain graph models, including Fuzzy, Intuitionistic Fuzzy, Neutrosophic, and Plithogenic Graphs, which are designed to address uncertainty in diverse applications. In this study, we introduce and analyze AT-Free Graphs, Claw-Free Graphs, and Triangle-Free Graphs within the framework of Fuzzy Graphs, investigating their properties and relationships in uncertain graph theory.
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22

Prabha, S. Celine, M. Palanivel, S. Amutha, N. Anbazhagan, Woong Cho, Hyoung-Kyu Song, Gyanendra Prasad Joshi, and Hyeonjoon Moon. "Solutions of Detour Distance Graph Equations." Sensors 22, no. 21 (November 2, 2022): 8440. http://dx.doi.org/10.3390/s22218440.

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Анотація:
Graph theory is a useful mathematical structure used to model pairwise relations between sensor nodes in wireless sensor networks. Graph equations are nothing but equations in which the unknown factors are graphs. Many problems and results in graph theory can be formulated in terms of graph equations. In this paper, we solved some graph equations of detour two-distance graphs, detour three-distance graphs, detour antipodal graphs involving with the line graphs.
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23

Ibarra, Sofía, and Luis Manuel Rivera. "The automorphism groups of some token graphs." Proyecciones (Antofagasta) 42, no. 6 (December 1, 2023): 1627–51. http://dx.doi.org/10.22199/issn.0717-6279-5954.

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Анотація:
The token graphs of graphs have been studied at least from the 80’s with different names and by different authors. The Johnson graph J(n, k) is isomorphic to the k-token graph of the complete graph Kn. To our knowledge, the unique results about the automorphism groups of token graphs are for the case of the Johnson graphs. In this paper we begin the study of the automorphism groups of token graphs of another graphs. In particular we obtain the automorphism group of the k-token graph of the path graph Pn, for n 6≠ 2k. Also, we obtain the automorphism group of the 2-token graph of the following graphs: cycle, star, fan and wheel graphs.
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24

K.R., Sridhara, та Mallikarjun Basanna Kattimani. "The (𝑎,𝑏)− Status Indices of Central Graphs of Some Standard Graph". International Journal of Basic Sciences and Applied Computing 8, № 8 (30 квітня 2022): 12–17. http://dx.doi.org/10.35940/ijbsac.h0476.048822.

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Анотація:
The sum of shortest distance between a vertex 𝒖 from all other vertices of a graph 𝑮 is called the index of the vertex 𝒖 and is denoted by 𝝈(𝒖). In this article, we have obtained, the(𝒂,𝒃)− status index [3] of central graphs of some standard graphs namely star graph, complete graph, cycle graph, wheel graph and friendship graph. Using this new index, we have computed 9 standard status indices of all these central graphs of standard graphs.
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25

Blinco, Andrew. "Theta graphs, graph decompositions and related graph labelling techniques." Bulletin of the Australian Mathematical Society 69, no. 1 (February 2004): 173–75. http://dx.doi.org/10.1017/s0004972700034377.

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26

Shim, Sooyeon, Junghun Kim, Kahyun Park, and U. Kang. "Accurate graph classification via two-staged contrastive curriculum learning." PLOS ONE 19, no. 1 (January 3, 2024): e0296171. http://dx.doi.org/10.1371/journal.pone.0296171.

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Анотація:
Given a graph dataset, how can we generate meaningful graph representations that maximize classification accuracy? Learning representative graph embeddings is important for solving various real-world graph-based tasks. Graph contrastive learning aims to learn representations of graphs by capturing the relationship between the original graph and the augmented graph. However, previous contrastive learning methods neither capture semantic information within graphs nor consider both nodes and graphs while learning graph embeddings. We propose TAG (Two-staged contrAstive curriculum learning for Graphs), a two-staged contrastive learning method for graph classification. TAG learns graph representations in two levels: node-level and graph level, by exploiting six degree-based model-agnostic augmentation algorithms. Experiments show that TAG outperforms both unsupervised and supervised methods in classification accuracy, achieving up to 4.08% points and 4.76% points higher than the second-best unsupervised and supervised methods on average, respectively.
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27

Zhang, Tong, Yun Wang, Zhen Cui, Chuanwei Zhou, Baoliang Cui, Haikuan Huang, and Jian Yang. "Deep Wasserstein Graph Discriminant Learning for Graph Classification." Proceedings of the AAAI Conference on Artificial Intelligence 35, no. 12 (May 18, 2021): 10914–22. http://dx.doi.org/10.1609/aaai.v35i12.17303.

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Анотація:
Graph topological structures are crucial to distinguish different-class graphs. In this work, we propose a deep Wasserstein graph discriminant learning (WGDL) framework to learn discriminative embeddings of graphs in Wasserstein-metric (W-metric) matching space. In order to bypass the calculation of W-metric class centers in discriminant analysis, as well as better support batch process learning, we introduce a reference set of graphs (aka graph dictionary) to express those representative graph samples (aka dictionary keys). On the bridge of graph dictionary, every input graph can be projected into the latent dictionary space through our proposed Wasserstein graph transformation (WGT). In WGT, we formulate inter-graph distance in W-metric space by virtue of the optimal transport (OT) principle, which effectively expresses the correlations of cross-graph structures. To make WGDL better representation ability, we dynamically update graph dictionary during training by maximizing the ratio of inter-class versus intra-class Wasserstein distance. To evaluate our WGDL method, comprehensive experiments are conducted on six graph classification datasets. Experimental results demonstrate the effectiveness of our WGDL, and state-of-the-art performance.
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28

Slilaty, Daniel. "Coloring permutation-gain graphs." Contributions to Discrete Mathematics 16, no. 1 (March 19, 2021): 47–52. http://dx.doi.org/10.55016/ojs/cdm.v16i1.62717.

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Анотація:
Correspondence colorings of graphs were introduced in 2018 by Dvořák and Postle as a generalization of list colorings of graphs which generalizes ordinary graph coloring. Kim and Ozeki observed that correspondence colorings generalize various notions of signed-graph colorings which again generalizes ordinary graph colorings. In this note we state how correspondence colorings generalize Zaslavsky's notion of gain-graph colorings and then formulate a new coloring theory of permutation-gain graphs that sits between gain-graph coloring and correspondence colorings. Like Zaslavsky's gain-graph coloring, our new notion of coloring permutation-gain graphs has well defined chromatic polynomials and lifts to colorings of the regular covering graph of a permutation-gain graph.
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29

Hanif, Muhammad Zeeshan, Naveed Yaqoob, Muhammad Riaz, and Muhammad Aslam. "Linear Diophantine fuzzy graphs with new decision-making approach." AIMS Mathematics 7, no. 8 (2022): 14532–56. http://dx.doi.org/10.3934/math.2022801.

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Анотація:
&lt;abstract&gt;&lt;p&gt;The concept of linear Diophantine fuzzy set (LDFS) is a new mathematical tool for optimization, soft computing, and decision analysis. The aim of this article is to extend the notion of graph theory towards LDFSs. We initiate the idea of linear Diophantine fuzzy graph (LDF-graph) as a generalization of certain theoretical concepts including, q-rung orthopair fuzzy graph, Pythagorean fuzzy graph, and intuitionistic fuzzy graph. We extend certain properties of crisp graph theory towards LDF-graph including, composition, join, and union of LDF-graphs. We elucidate these operations with various illustrations. We analyze some interesting results that the composition of two LDF-graphs is a LDF-graph, cartesian product of two LDF-graphs is a LDF-graph, and the join of two LDF-graphs is a LDF-graph. We describe the idea of homomorphisms for LDF-graphs. We observe the equivalence relation via an isomorphism between LDF-graphs. Some significant results related to complement of LDF-graph are also investigated. Lastly, an algorithm based on LDFSs and LDF-relations is proposed for decision-making problems. A numerical example of medical diagnosis application is presented based on proposed approach.&lt;/p&gt;&lt;/abstract&gt;
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30

Sridhara, K.R., та Mallikarjun Basanna Kattimani Dr. "The (𝑎,𝑏)− Status Indices of Central Graphs of Some Standard Graph". International Journal of Basic Sciences and Applied Computing (IJBSAC) 8, № 8 (30 квітня 2022): 12–17. https://doi.org/10.35940/ijbsac.H0476.048822.

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Анотація:
<strong>Abstract:</strong> The sum of shortest distance between a vertex 𝒖 from all other vertices of a graph 𝑮 is called the index of the vertex 𝒖 and is denoted by 𝝈(𝒖). In this article, we have obtained, the(𝒂,𝒃)&minus; status index [3] of central graphs of some standard graphs namely star graph, complete graph, cycle graph, wheel graph and friendship graph. Using this new index, we have computed 9 standard status indices of all these central graphs of standard graphs.
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31

Ismail, Sumarno, Isran K. Hasan, Tesya Sigar, and Salmun K. Nasib. "RAINBOW CONNECTION NUMBER AND TOTAL RAINBOW CONNECTION NUMBER OF AMALGAMATION RESULTS DIAMOND GRAPH(〖Br〗_4) AND FAN GRAPH(F_3)." BAREKENG: Jurnal Ilmu Matematika dan Terapan 16, no. 1 (March 21, 2022): 023–30. http://dx.doi.org/10.30598/barekengvol16iss1pp023-030.

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Анотація:
If be a graph and edge coloring of G is a function , rainbow connection number is the minimum-k coloration of the rainbow on the edge of graph G and denoted by rc(G). Rainbow connection numbers can be applied to the result of operations on some special graphs, such as diamond graphs and fan graphs. Graph operation is a method used to obtain a new graph by combining two graphs. This study performed amalgamation operations to obtain rainbow connection numbers and rainbow-total-connection numbers in diamond graphs ( ) and fan graphs ( ) or . Based on the research, it is obtained that the rainbow-connection number theorem on the amalgamation result of the diamond graph ( ) and fan graph ( is with . Furthermore, the theorem related to the total rainbow-connection number on the amalgamation result of the diamond graph( ) and the fan graph ( is obtained, namely with .
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32

K, Radha, and Jasmine Kingsly A. "Cardinality and Isomorphic Properties of Hajós Graphs and Hajós Fuzzy Graphs." Indian Journal of Science and Technology 17, no. 37 (September 28, 2024): 3871–80. https://doi.org/10.17485/IJST/v17i37.2492.

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Анотація:
Abstract <strong>Objectives:</strong>&nbsp;Fuzzy graphs allow uncertainty in the ideas characterizing vertices and edges to be included when modelling real-world scenarios into graph models. This study aims to introduce Haj&oacute;s fuzzy graph, cardinality of Haj&oacute;s graph and Haj&oacute;s fuzzy graph and to explore many properties.<strong>&nbsp;Methods :</strong>&nbsp;Here, the Haj&oacute;s construction is applied to two fuzzy graphs, and the Haj&oacute;s fuzzy graph is defined by assigning membership values to all the edges and vertices of the newly formed fuzzy graph. Fuzzy graph conditions are verified for the assigned membership values. Examples are used to illustrate the concept.&nbsp;<strong>Findings:</strong>&nbsp;The Haj&oacute;s fuzzy graph's size and order are established. Isomorphic property is discussed. The number of Haj&oacute;s (fuzzy) graphs that exist for some combinations of two (fuzzy) graphs is calculated.&nbsp;<strong>Novelty:</strong>&nbsp;Based on Haj&oacute;s construction, the Haj&oacute;s fuzzy graph is defined. Based on the fact that different choices of vertices and edges produce different Haj&oacute;s (fuzzy) graph, its cardinality is defined. The notion of cardinality is a novel concept in Hajos graph which helps to find the total number of Haj&oacute;s (fuzzy) graphs. The cardinality of the Haj&oacute;s (fuzzy) graph which arises from two (fuzzy) graphs that can be a cycle, path, regular graph, complete graph and complete bipartite graph are derived. Mathematics subject classification. (2010): 05C72, 05C76 <strong>Keywords:</strong> Haj&oacute;s Graph, Haj&oacute;s fuzzy graph, Order, Size, Isomorphism, Cardinality of Haj&oacute;s fuzzy graph
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33

Tamanna, Kumari, Kumar Kamal, and Agarwal Richa. "A STUDY OF STRUCTURE OF GRAPH INVARIANT WITH THEIR INDEPENDENCE NUMBERS AND CAYLEY GRAPHS." Soch – Mastnath Journal Of Science & Technology 17, no. 2 (September 30, 2022): 1–8. https://doi.org/10.5281/zenodo.8009349.

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Анотація:
ABSTRACT In this paper the independence numbers and algebraic properties of graph invariant has been discussed and studied. The graph invariant is generalization of Petersen graphs whose independence numbers have been previously known. The authors studied the bounds for the independence number of different graph invariant and sub-classes of graph invariant, and exactly determine the independence number for other graph invariant and sub-classes of graph invariant. Authors also analysed the automorphism groups of the graph invariant.
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34

SINHA, DEEPA, and DEEPAKSHI SHARMA. "On Square and 2-path Signed Graph." Journal of Interconnection Networks 16, no. 01 (March 2016): 1550011. http://dx.doi.org/10.1142/s0219265915500115.

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Анотація:
A signed graph is an ordered pair [Formula: see text], where [Formula: see text] is a graph G = (V, E), called the underlying graph of S and [Formula: see text] is a function from the edge set E of Su into the set {+, -}, called the signature of S. In this paper, we characterize all those signed graphs whose 2-path signed graphs are isomorphic to their square signed graph along with algorithm to check the same. In other sections we find the characterization of signed graph S such that [Formula: see text] where D is a derived signed graph of the signed graph S such as: line signed graphs, total signed graphs, common edge signed graphs, splitting signed graphs. Also each characterization is supported by algorithms for the same.
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35

KAPOOR, SANJIV, and XIANG-YANG LI. "PROXIMITY STRUCTURES FOR GEOMETRIC GRAPHS." International Journal of Computational Geometry & Applications 20, no. 04 (August 2010): 415–29. http://dx.doi.org/10.1142/s0218195910003360.

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Анотація:
In this paper we study proximity graph structures like Delaunay triangulations based on geometric graphs, i.e. graphs which are subgraphs of the complete geometric graph. Given an arbitrary geometric graph G, we define Voronoi diagrams, Delaunay triangulations, relative neighborhood graphs, Gabriel graphs which are related to the graph structure and then study their complexities when G is a general geometric graph or G is some special graph derived from the application area of wireless networks. Besides being of fundamental interest these structures have applications in topology control for wireless networks.
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36

Manash Protim Borah. "Quasi- Laplacian energy of some novel classes of graphs." Communications on Applied Nonlinear Analysis 32, no. 7s (January 23, 2025): 530–41. https://doi.org/10.52783/cana.v32.3462.

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Анотація:
We formulate the relationship of quasi-Laplacian energy of some novel classes of graphs with their corresponding original graphs. The novel graphs in our discussion are the -graph, -graph, -graph, total graph, and their join and corona operations graphs. The whole formulation is based on the relationship between quasi-Laplacian energy and the vertex degrees of the novel graph. It is also noted that quasi-Laplacian energy is closely related with the first Zagreb index, number of vertices and edges of the graph. The exact formulas of quasi-Laplacian energy of novel graphs are obtained in terms of the corresponding quasi-Laplacian energies, the first Zagreb indices, and the number of vertices and edges of the original graphs.
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37

Keerthi, G. Mirajkar* Bhagyashri R. Doddamani Priyanka Y. B. "THE REFORMULATED FIRST ZAGREB INDEX OF THE LINE GRAPHS OF THE SUBDIVISION GRAPH FOR CLASS OF GRAPHS." INTERNATIONAL JOURNAL OF ENGINEERING SCIENCES & RESEARCH TECHNOLOGY 5, no. 10 (October 5, 2016): 144–49. https://doi.org/10.5281/zenodo.159334.

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Анотація:
The reformulated first Zagreb index is the edge version of first Zagreb index of chemical graph theory. The aim of this paper is to obtain an expression for the reformulated first Zagreb index of the some class of graphs such as Tadpole graph, Wheel graph, Ladder graph. Further we also obtain the reformulated first Zagreb index of the line graph, subdivision graph and line graph of subdivision graph for class of graphs.
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38

Ling, Yawen, Jianpeng Chen, Yazhou Ren, Xiaorong Pu, Jie Xu, Xiaofeng Zhu, and Lifang He. "Dual Label-Guided Graph Refinement for Multi-View Graph Clustering." Proceedings of the AAAI Conference on Artificial Intelligence 37, no. 7 (June 26, 2023): 8791–98. http://dx.doi.org/10.1609/aaai.v37i7.26057.

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Анотація:
With the increase of multi-view graph data, multi-view graph clustering (MVGC) that can discover the hidden clusters without label supervision has attracted growing attention from researchers. Existing MVGC methods are often sensitive to the given graphs, especially influenced by the low quality graphs, i.e., they tend to be limited by the homophily assumption. However, the widespread real-world data hardly satisfy the homophily assumption. This gap limits the performance of existing MVGC methods on low homophilous graphs. To mitigate this limitation, our motivation is to extract high-level view-common information which is used to refine each view's graph, and reduce the influence of non-homophilous edges. To this end, we propose dual label-guided graph refinement for multi-view graph clustering (DuaLGR), to alleviate the vulnerability in facing low homophilous graphs. Specifically, DuaLGR consists of two modules named dual label-guided graph refinement module and graph encoder module. The first module is designed to extract the soft label from node features and graphs, and then learn a refinement matrix. In cooperation with the pseudo label from the second module, these graphs are refined and aggregated adaptively with different orders. Subsequently, a consensus graph can be generated in the guidance of the pseudo label. Finally, the graph encoder module encodes the consensus graph along with node features to produce the high-level pseudo label for iteratively clustering. The experimental results show the superior performance on coping with low homophilous graph data. The source code for DuaLGR is available at https://github.com/YwL-zhufeng/DuaLGR.
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39

Li, Qiuping, Liangwen Tang, Qingyun Liu, and Mugang Lin. "A New Family of Multipartition Graph Operations and Its Applications in Constructing Several Special Graphs." Symmetry 17, no. 3 (March 20, 2025): 467. https://doi.org/10.3390/sym17030467.

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Анотація:
A new family of graph operations based on multipartite graph with an arbitrary number of parts is defined and their applications are explored in this paper. The complete spectra of graphs derived from multipartite graphs are determined. Because the adjacency matrix of the multipartite graph is symmetric, we can use it to generate an unlimited number of special symmetric graphs. Methods for generating countless new families of integral graphs using these multipartite graph operations have been presented. By applying these multipartite graph operations, we can construct infinitely many orderenergetic graphs from orderenergetic or non-orderenergetic graphs. Additionally, infinite pairs of equienergetic and non-cospectral graphs can be generated through these new operations. Moreover, this kind of graph operation can also be used to construct other special graphs related to eigenvalues and energy.
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40

Ali, Muhammad Asad, Muhammad Shoaib Sardar, Imran Siddique, and Dalal Alrowaili. "Vertex-Based Topological Indices of Double and Strong Double Graph of Dutch Windmill Graph." Journal of Chemistry 2021 (October 26, 2021): 1–12. http://dx.doi.org/10.1155/2021/7057412.

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Анотація:
A measurement of the molecular topology of graphs is known as a topological index, and several physical and chemical properties such as heat formation, boiling point, vaporization, enthalpy, and entropy are used to characterize them. Graph theory is useful in evaluating the relationship between various topological indices of some graphs derived by applying certain graph operations. Graph operations play an important role in many applications of graph theory because many big graphs can be obtained from small graphs. Here, we discuss two graph operations, i.e., double graph and strong double graph. In this article, we will compute the topological indices such as geometric arithmetic index GA , atom bond connectivity index ABC , forgotten index F , inverse sum indeg index ISI , general inverse sum indeg index ISI α , β , first multiplicative-Zagreb index PM 1 and second multiplicative-Zagreb index PM 2 , fifth geometric arithmetic index GA 5 , fourth atom bond connectivity index ABC 4 of double graph, and strong double graph of Dutch Windmill graph D 3 p .
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41

Kaladevi, V., R. Murugesan, and K. Pattabiraman. "First reformulated Zagreb indices of some classes of graphs." Carpathian Mathematical Publications 9, no. 2 (January 2, 2018): 134–44. http://dx.doi.org/10.15330/cmp.9.2.134-144.

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Анотація:
A topological index of a graph is a parameter related to the graph; it does not depend on labeling or pictorial representation of the graph. Graph operations plays a vital role to analyze the structure and properties of a large graph which is derived from the smaller graphs. The Zagreb indices are the important topological indices found to have the applications in Quantitative Structure Property Relationship(QSPR) and Quantitative Structure Activity Relationship(QSAR) studies as well. There are various study of different versions of Zagreb indices. One of the most important Zagreb indices is the reformulated Zagreb index which is used in QSPR study.&#x0D; In this paper, we obtain the first reformulated Zagreb indices of some derived graphs such as double graph, extended double graph, thorn graph, subdivision vertex corona graph, subdivision graph and triangle parallel graph. In addition, we compute the first reformulated Zagreb indices of two important transformation graphs such as the generalized transformation graph and generalized Mycielskian graph.
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42

T. K., Jahfar, and Chithra A. V. "Energy and Randić energy of special graphs." Proyecciones (Antofagasta) 41, no. 4 (July 25, 2022): 855–77. http://dx.doi.org/10.22199/issn.0717-6279-4616.

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Анотація:
In this paper, we determine the Randić energy of the m-splitting graph, the m-shadow graph and the m-duplicate graph of a given graph, m being an arbitrary integer. Our results allow the construction of an infinite sequence of graphs having the same Randić energy. Further, we determine some graph invariants like the degree Kirchhoff index, the Kemeny’s constant and the number of spanning trees of some special graphs. From our results, we indicate how to obtain infinitely many pairs of equienergetic graphs, Randić equienergetic graphs and also, infinite families of integral graphs.
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43

Firmansah, Fery, and Wed Giyarti. "Odd harmonious labeling on the amalgamation of the generalized double quadrilateral windmill graph." Desimal: Jurnal Matematika 4, no. 3 (November 30, 2021): 373–78. http://dx.doi.org/10.24042/djm.v4i3.10823.

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Анотація:
Graph labeling is one of the topics of graph theory that is growing very rapidly both in terms of theory and application. A graph that satisfies the labeling property of odd harmonious is called an odd harmonious graph. The method used in this research is qualitative research by developing a theory and a new class of graphs from odd harmonious graphs. In this research, a new graph class construction will be given in the form of an amalgamation of the generalized double quadrilateral windmill graph. Furthermore, it will be proved that the amalgamation of the generalized double quadrilateral windmill graph is an odd harmonious graph. So that the results of the research show that the amalgamation of the generalized double quadrilateral windmill graph is a new graph class of odd harmonious graphs.
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44

Pattabiraman, K., and P. Kandan. "Weighted PI index of corona product of graphs." Discrete Mathematics, Algorithms and Applications 06, no. 04 (October 10, 2014): 1450055. http://dx.doi.org/10.1142/s1793830914500554.

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Анотація:
In this paper, we present exact formula for the weighted PI index of corona product of two connected graphs in terms of other graph invariants including the PI index, first Zagreb index and second Zagreb index. Then, we apply our result to compute the weighted PI indices of t-fold bristled graph, bottleneck graph, sunlet graph, star graph, fan graph, wheel graph and some classes of bridge graphs.
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45

Singh, Priyanka, and Pratima Panigrahi. "On Self-Centeredness of Product of Graphs." International Journal of Combinatorics 2016 (August 7, 2016): 1–4. http://dx.doi.org/10.1155/2016/2508156.

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Анотація:
A graph G is said to be a self-centered graph if the eccentricity of every vertex of the graph is the same. In other words, a graph is a self-centered graph if radius and diameter of the graph are equal. In this paper, self-centeredness of strong product, co-normal product, and lexicographic product of graphs is studied in detail. The necessary and sufficient conditions for these products of graphs to be a self-centered graph are also discussed. The distance between any two vertices in the co-normal product of a finite number of graphs is also computed analytically.
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46

A. Antony mary, A., A. Amutha, and M. S. Franklin Thamil Selvi. "A Study on Slope Number of Certain Classes of Bipartite Graphs." International Journal of Engineering & Technology 7, no. 4.10 (October 2, 2018): 440. http://dx.doi.org/10.14419/ijet.v7i4.10.21036.

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Анотація:
Graph drawing is the most important area of mathematics and computer science which combines methods from geometric graph theory and information visualization. Generally, graphs are represented to explore some intellectual ideas. Graph drawing is the familiar concept of graph theory. It has many quality measures and one among them is the slope number. Slope number problem is an optimization problem and is NP-hard to determine the slope number of any arbitrary graph. In the present paper, the investigation on slope number of bipartite graph is studied elaborately. Since the bipartite graphs creates one of the most intensively investigated classes of graphs, we consider few classes of graphs and discussed structural behavior of such graphs.
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47

Pramanik, L. K. "Centers in inserted graphs." Filomat 21, no. 2 (2007): 21–30. http://dx.doi.org/10.2298/fil0702021p.

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Анотація:
In this paper we study some concepts involving distance in inserted graphs with an emphasize on centers in inserted graphs. More precisely we prove that for every non-trivial connected graph H there exists a graph G such that H is the center of G and the inserted graph of H is the center of the inserted graph of G. Graphs which are the periphery of some inserted graph are characterized. .
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48

Coutino, Mario, Sundeep Prabhakar Chepuri, Takanori Maehara, and Geert Leus. "Fast Spectral Approximation of Structured Graphs with Applications to Graph Filtering." Algorithms 13, no. 9 (August 31, 2020): 214. http://dx.doi.org/10.3390/a13090214.

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Анотація:
To analyze and synthesize signals on networks or graphs, Fourier theory has been extended to irregular domains, leading to a so-called graph Fourier transform. Unfortunately, different from the traditional Fourier transform, each graph exhibits a different graph Fourier transform. Therefore to analyze the graph-frequency domain properties of a graph signal, the graph Fourier modes and graph frequencies must be computed for the graph under study. Although to find these graph frequencies and modes, a computationally expensive, or even prohibitive, eigendecomposition of the graph is required, there exist families of graphs that have properties that could be exploited for an approximate fast graph spectrum computation. In this work, we aim to identify these families and to provide a divide-and-conquer approach for computing an approximate spectral decomposition of the graph. Using the same decomposition, results on reducing the complexity of graph filtering are derived. These results provide an attempt to leverage the underlying topological properties of graphs in order to devise general computational models for graph signal processing.
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49

Rice, Sean J., and Douglas J. Gillan. "Effects of Graphs on Text Comprehension." Proceedings of the Human Factors and Ergonomics Society Annual Meeting 44, no. 21 (July 2000): 3–443. http://dx.doi.org/10.1177/154193120004402117.

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Анотація:
Sources of information, from textbooks to multimedia encyclopedias combine text and graphics. This paper applies the construction-integration model of text comprehension to the processing of text and quantitative graphs. The model suggests that, when reading a graph that has information that is redundant with the text, a reader may directly connect the propositional information from the graph and the text due to overlap. When reading a graph that has information that elaborates on the text, a reader makes inferences to connect the propositional information from the graph and the text. Hypotheses from this model are that (1) use of an elaborative graph should produce better comprehension if the graph and text are concurrent, whereas (2) use of a redundant graph should produce better comprehension if the graph and text are separated. Three experiments tested these hypotheses by giving readers text with three types of graphs – text only, redundant graphs, and elaborative graphs — with the graphs either concurrent with (Experiment 1), before (Experiment 2A), or after (Experiment 2B) the text, then giving readers text-based or inference-based questions. In all three orders of the graph and text, performance on the inference-based questions was better in the elaborative graph condition than in the redundant graph condition. The discussion extends the application of the construction-integration model to reading text with graphs.
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50

Prasetyo, Joko. "FAKTORISASI PADA GRAF REGULER." EDUPEDIA 4, no. 1 (April 18, 2020): 75. http://dx.doi.org/10.24269/ed.v4i1.434.

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Анотація:
This research aims to: (1) know the criteria of a graph that has a -factor, (2) know the conditions of a regular graph that has a 1-factorization , (3) know the conditions of a regular graph that has a 2-factorization.This research is a qualitative descriptive study using the method of literature study or literature review where a study of books, scientific journals, and other literature languages is carried out relating to factorization on regular graphs. This research begins by discussing the definitions and examples of euler graphs and regular bipartite multigraphs. Next in reviewing the terms of a regular graph which has a 1-factorization and which has a 2-factorization, it starts by discussing the definition and theorem of matching on bipartite graphs, definitions and examples of factorization graphs, then discussing the proof of theorem of regular graphs that have a 1-factor and a regular graph which has a 2-factor.The results of this study indicate that: (1) Graph is said to be -factorable or can be factored into -factor , if can be decomposed or be eksplained into spanning subgraphs , where each has a -factor and is edge-disjoint from , that is 1) 2) … n) = . (2) The condition for a graph that has a 1-factorization is, if the graph is a -regular bipartite multigraph, with . (3) The condition for a graph that has a 2-factorization is, if the graph is a -regular graph, with . Key words: Bipartite graphs, Factorization, Decomposition, Regular graph.
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