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Journal articles on the topic 'Graph theory'

Author: Grafiati

Published: 4 June 2021

Last updated: 9 June 2025

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1

Nada A Laabi. "Subring in Graph Theory." Advances in Nonlinear Variational Inequalities 27, no. 4 (September 5, 2024): 284–87. http://dx.doi.org/10.52783/anvi.v27.1526.

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In our study, we delved into the intricacies of graph theory by exploring the properties of subrings within various types of graphs. By focusing on prime graphs and simple graphs, we unraveled the complex relationship between subring-prime graphs. Additionally, we delved into the concept of homomorphism within both simple subring graphs and prime subring graphs, adding depth to our analysis.

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Aharwal, Ramesh Prasad. "Graph Theory Applications in Machine Learning." International Journal for Research in Applied Science and Engineering Technology 13, no. 3 (March 31, 2025): 645–48. https://doi.org/10.22214/ijraset.2025.67337.

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Graph theory is the Branch of Discrete mathematics which plays a key role in Machine Learning and Data Science. Graph Theory in Machine Learning states to the application of mathematical structures known as graphs to model pairwise relations between objects in machine learning. A graph in this framework is a set of objects, called nodes, connected by links, known as edges. Each edge may be directed or undirected. In mathematics, graph theory is one of the important fields used in structural models. This paper explores the applications of Graph theory and various types of graphs in Machine Learning. The paper also discusses the advantages and various types of graph theory Algorithms which are used in Machine learning.

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ILYUTKO, DENIS PETROVICH, and VASSILY OLEGOVICH MANTUROV. "INTRODUCTION TO GRAPH-LINK THEORY." Journal of Knot Theory and Its Ramifications 18, no. 06 (June 2009): 791–823. http://dx.doi.org/10.1142/s0218216509007191.

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The present paper is an introduction to a combinatorial theory arising as a natural generalization of classical and virtual knot theory. There is a way to encode links by a class of "realizable" graphs. When passing to generic graphs with the same equivalence relations we get "graph-links". On one hand graph-links generalize the notion of virtual link, on the other hand they do not detect link mutations. We define the Jones polynomial for graph-links and prove its invariance. We also prove some a generalization of the Kauffman–Murasugi–Thistlethwaite theorem on "minimal diagrams" for graph-links.

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4

C.S., Harisha. "Graph Theory Approach to Number Theory Theorems." Journal of Advanced Research in Dynamical and Control Systems 12, no. 01-Special Issue (February 13, 2020): 568–72. http://dx.doi.org/10.5373/jardcs/v12sp1/20201105.

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5

Rohini Gore, Tejal Gore, Namrata Rokade, and Yogesh Mandlik. "“Applications of Graph Theory”." International Journal of Latest Technology in Engineering Management & Applied Science 14, no. 3 (April 4, 2025): 148–50. https://doi.org/10.51583/ijltemas.2025.140300018.

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Abstract: Graph theory is a fundamental area of mathematics with diverse applications across multiple fields. It provides a structural framework for solving complex problems by representing objects and their relationships as graphs. In computer science, graph theory is used in networking, algorithms, and artificial intelligence. It plays a crucial role in transportation and logistics, optimizing routes and traffic flow. Social networks leverage graph models for community detection and influence analysis. In biology and medicine, graph theory aids in understanding neural connections, disease spread, and genomic structures. Additionally, it is applied in cyber security, linguistics, operations research, and chemistry. The versatility of graph theory makes it an essential tool for analysing and solving real-world problems efficiently.

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Möhring, Rolf H. "Algorithmic graph theory and perfect graphs." Order 3, no. 2 (June 1986): 207–8. http://dx.doi.org/10.1007/bf00390110.

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Fujita, Takaaki. "Review of Rough Turiyam Neutrosophic Directed Graphs and Rough Pentapartitioned Neutrosophic Directed Graphs." Neutrosophic Optimization and Intelligent Systems 5 (March 4, 2025): 48–79. https://doi.org/10.61356/j.nois.2025.5500.

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Graph theory, a fundamental branch of mathematics, examines relationships between entities through the use of vertices and edges. Within this field, Uncertain Graph Theory has developed as a powerful framework to represent the uncertainties found in real world networks. Among the various uncertain graph models, Turiyam Neutrosophic Graphs and Pentapartitioned Neutrosophic Graphs are well-established. However, their extension to Directed Graphs remains relatively unexplored. To address this gap, this paper presents the concepts of the Turiyam Neutrosophic Directed Graph and the Pentapartitioned Neutrosophic Directed Graph, thereby broadening the scope of uncertain graph theory.

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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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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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Bumby, Richard T., and Dana May Latch. "Categorical constructions in graph theory." International Journal of Mathematics and Mathematical Sciences 9, no. 1 (1986): 1–16. http://dx.doi.org/10.1155/s0161171286000017.

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This paper presents some graph-theoretic questions from the viewpoint of the portion of category theory which has become common knowledge. In particular, the reader is encouraged to consider whether there is only one natural category of graphs and how theories of directed graphs and undirected graphs are related.

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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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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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Pardo-Guerra, Sebastian, Vivek Kurien George, Vikash Morar, Joshua Roldan, and Gabriel Alex Silva. "Extending Undirected Graph Techniques to Directed Graphs via Category Theory." Mathematics 12, no. 9 (April 29, 2024): 1357. http://dx.doi.org/10.3390/math12091357.

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We use Category Theory to construct a ‘bridge’ relating directed graphs with undirected graphs, such that the notion of direction is preserved. Specifically, we provide an isomorphism between the category of simple directed graphs and a category we call ‘prime graphs category’; this has as objects labeled undirected bipartite graphs (which we call prime graphs), and as morphisms undirected graph morphisms that preserve the labeling (which we call prime graph morphisms). This theoretical bridge allows us to extend undirected graph techniques to directed graphs by converting the directed graphs into prime graphs. To give a proof of concept, we show that our construction preserves topological features when applied to the problems of network alignment and spectral graph clustering.

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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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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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Seema Varghese and Brinta Babu. "An overview on graph products." International Journal of Science and Research Archive 10, no. 1 (October 30, 2023): 966–71. http://dx.doi.org/10.30574/ijsra.2023.10.1.0848.

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Graph product is a very basic idea in graph theory. Graph products include a wide range of operations that join two or more existing graphs to produce new graphs with distinctive properties and uses. In this paper, we explore four forms of graph products, their characteristics, and their significance within the broader framework of graph theory.

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Cvetkovic, Dragos, and Slobodan Simic. "Towards a spectral theory of graphs based on the signless Laplacian, I." Publications de l'Institut Math?matique (Belgrade) 85, no. 99 (2009): 19–33. http://dx.doi.org/10.2298/pim0999019c.

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A spectral graph theory is a theory in which graphs are studied by means of eigenvalues of a matrix M which is in a prescribed way defined for any graph. This theory is called M-theory. We outline a spectral theory of graphs based on the signless Laplacians Q and compare it with other spectral theories, in particular with those based on the adjacency matrix A and the Laplacian L. The Q-theory can be composed using various connections to other theories: equivalency with A-theory and L-theory for regular graphs, or with L-theory for bipartite graphs, general analogies with A-theory and analogies with A-theory via line graphs and subdivision graphs. We present results on graph operations, inequalities for eigenvalues and reconstruction problems.

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15

V.R.Kulli. "NEW DIRECTION IN THE THEORY OF GRAPH INDEX IN GRAPHS." INTERNATIONAL JOURNAL OF ENGINEERING SCIENCES & RESEARCH TECHNOLOGY 11, no. 12 (January 5, 2023): 1–8. https://doi.org/10.5281/zenodo.7505790.

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Since 1972, several graph indices were introduced and studied. In this paper, we define the Banhatti degree of vertex in a graph. We propose the first and second E-Banhatti indices of a graph. A study of E-Banhatti indices in Mathematical Chemistry is a New Direction in the Theory of Graph Index in Graphs. Also we compute these newly defined E-Banhatti indices and their corresponding exponentials for wheel graphs, friendship graphs and some important nanostructures which are appeared in nanoscience.

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Valdiviezo, Manuel Baro Judith, Mayra Perez, and Rosa Ma. Toral. "Metanalysis of Algebraic Graph Theory." International Journal of Research Publication and Reviews 6, no. 2 (February 2025): 590–496. https://doi.org/10.55248/gengpi.6.0225.0721.

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17

Alrawayati, Hawa Ahmed, and Ümit Tokeşer. "Spectral Integral Variation of Graph Theory." Asian Journal of Mathematics and Computer Research 32, no. 2 (March 11, 2025): 151–60. https://doi.org/10.56557/ajomcor/2025/v32i29173.

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Spectral integral variation in graph theory explores the interplay between the spectral properties of graphs and their topological and geometrical characteristics. This study focuses on the eigenvalues and eigenvectors of graph-related matrices, such as the adjacency matrix and the Laplacian matrix, and their implications for understanding graph structure, connectivity, and dynamics. By examining integral variations, we establish a framework for analyzing how spectral properties change under perturbations, such as edge weight modifications and graph transformations. This paper discusses the significance of cographs as a specific class of graphs that exhibit robust spectral characteristics, highlighting their linear independence and absence of certain induced subgraphs. Applications in network design, clustering, and dynamic systems are presented, demonstrating the utility of spectral methods in real-world scenarios. This work aims to bridge discrete and continuous perspectives in graph theory, providing a comprehensive analysis of spectral variations and their implications for both theoretical research and practical applications.

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Hinz, Andreas M. "Graph Theory of Tower Tasks." Behavioural Neurology 25, no. 1 (2012): 13–22. http://dx.doi.org/10.1155/2012/743238.

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The appropriate mathematical model for the problem space of tower transformation tasks is the state graph representing positions of discs or balls and their moves. Graph theoretical quantities like distance, eccentricities or degrees of vertices and symmetries of graphs support the choice of problems, the selection of tasks and the analysis of performance of subjects whose solution paths can be projected onto the graph. The mathematical model is also at the base of a computerized test tool to administer various types of tower tasks.

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19

Rodriguez, Jose. "Graph Theory." Symmetry 10, no. 1 (January 22, 2018): 32. http://dx.doi.org/10.3390/sym10010032.

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Lloyd, E. Keith, and W. T. Tutte. "Graph Theory." Mathematical Gazette 69, no. 447 (March 1985): 69. http://dx.doi.org/10.2307/3616480.

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Anderson, Ian, and Reinhard Diestel. "Graph Theory." Mathematical Gazette 85, no. 502 (March 2001): 176. http://dx.doi.org/10.2307/3620535.

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Abbott, Steve, and W. T. Tutte. "Graph Theory." Mathematical Gazette 85, no. 504 (November 2001): 559. http://dx.doi.org/10.2307/3621812.

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23

Diestel, Reinhard, Robin Thomas, and Gábor Tardos. "Graph Theory." Oberwolfach Reports 10, no. 1 (2013): 67–128. http://dx.doi.org/10.4171/owr/2013/02.

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Diestel, Reinhard, Daniel Král', and Paul Seymour. "Graph Theory." Oberwolfach Reports 13, no. 1 (2016): 51–86. http://dx.doi.org/10.4171/owr/2016/2.

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25

Geelen, Jim, Daniel Král', and Alexander Scott. "Graph Theory." Oberwolfach Reports 16, no. 1 (February 26, 2020): 5–63. http://dx.doi.org/10.4171/owr/2019/1.

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26

Gupta, Barkha. "Graph Theory." International Journal of Applied Research on Information Technology and Computing 12, no. 1to3 (2021): 31–45. http://dx.doi.org/10.5958/0975-8089.2021.00006.3.

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Geelen, Jim, Daniel Král', and Alex Scott. "Graph Theory." Oberwolfach Reports 19, no. 1 (March 10, 2023): 5–78. http://dx.doi.org/10.4171/owr/2022/1.

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Vidyashree H. R. "Some Derived Graphs of Ananta-Graphs." Panamerican Mathematical Journal 35, no. 3s (January 20, 2025): 522–27. https://doi.org/10.52783/pmj.v35.i3s.4246.

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Graph theory provides powerful tools for analyzing mathematical structures and sequences. The Ananta-graph, derived from the Collatz conjecture, represents integer transformations through directed edges, capturing number relationships under n→3n+1 and n→n/2 operations. This paper explores several derived graphs from the Ananta-graph, including line, middle, Mycielskian, subdivision, total, core, power, splitting and kernel graph, analyzing their structural properties and mathematical significance. By analyzing these derived graphs, we provide deeper insights into the topological, algebraic and computational properties of the Ananta-graph, contributing to both graph theory and number theory applications. This research further provides new perspectives on integer sequences and transformation-based graph models.

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Pradeep Jha, Rajeshri Prajapati, Amit Parikh,. "Special Graphs of Euler’s Family* and Tracing Algorithm- (A New Approach)." Proceeding International Conference on Science and Engineering 11, no. 1 (February 18, 2023): 2243–51. http://dx.doi.org/10.52783/cienceng.v11i1.400.

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There are in graph theory, some known graphs which date back from centuries. [Euler graph, Hamiltonian graph etc.] These graphs are basic roots for development of graph theory. In this paper we have discussed the novel concept of tracing Euler tour. It depends on the concept of Link vertex - a join vertex of finite number of cycles as components of Euler graph. In addition to this, a new notion of isomorphic transformation of given graph on to a given line segment known as ‘Linear Graph’ also plays an important role for tracing the Euler graph.

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kumar, Nand kishor, and Priti Singh. "Symmetry in the context of the strongly ⋆- graph, Cube Difference Labeling graph, Triangular Snake graph, and Theta graph." Journal of Scientific Research 67, no. 04 (2023): 52–57. http://dx.doi.org/10.37398/jsr.2023.670409.

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This paper presents the findings of a brief history of graph theory as well as an outline of the theory itself. This page discusses the strongly-graph, the cube difference labeling graph, the triangular snake graph, and the theta graph. In addition, we define them, present a formula, and explain the symmetrical relationship that exists between these graphs. Include some new graph families in your explanation, as well as examples and drawings.

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SAVITA S SHINDE, VIJAY. M.P, and SHIVAKUMAR MD. "Exploring Graph Theory For Practical Solutions." Innovative Research Thoughts 4, no. 8 (November 30, 2018): 85–94. https://doi.org/10.36676/irt.v4.i8.1556.

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Graph theory is widely used to prove many mathematical theorems and models. This paper present the various applications and techniques of graph theory to solve problems in different fields of science and technology in addition to mathematics. A graph can be used to represent almost any physical situation involving discrete objects and a relationship among them. This abstract provides a concise overview of graph theory’s foundational principles, including graph types(such as directed, undirected, weighted, and unweighted graphs), basics terminologies( vertices, edges, paths, cycles), and essential theorems (e.g, Euler’s theorem Hamiltonian cycles). Moreover, it highlights practical applications of graph theory, such as shortest path algorithms (e.g., Dijkstra’s algorithm), network flow optimization, and graph coloring problems. By unraveling the intricacies of graph theory, this abstract aims to foster a deeper understanding of its role in shaping modern computational paradigms and problem-solving methodologies. These fields include website design, chemistry, biology, computer science, software engineering & operations research.

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Yurttas Gunes, Aysun, Muge Togan, Musa Demirci, and Ismail Naci Cangul. "Harmonic Index and Zagreb Indices of Vertex-Semitotal Graphs." European Journal of Pure and Applied Mathematics 13, no. 5 (December 27, 2020): 1260–69. http://dx.doi.org/10.29020/nybg.ejpam.v13i5.3725.

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Graph theory is one of the rising areas in mathematics due to its applications in many areas of science. Amongst several study areas in graph theory, spectral graph theory and topological descriptors are in front rows. These descriptors are widely used in QSPR/QSAR studies in mathematical chemistry. Vertex-semitotal graphs are one of the derived graph classes which are useful in calculating several physico-chemical properties of molecular structures by means of molecular graphs modelling the molecules. In this paper, several topological descriptors of vertex-semitotal graphs are calculated. Some new relations on these values are obtained by means of a recently defined graph invariant called omega invariant.

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VISHVESHWARA, SARASWATHI, K. V. BRINDA, and N. KANNAN. "PROTEIN STRUCTURE: INSIGHTS FROM GRAPH THEORY." Journal of Theoretical and Computational Chemistry 01, no. 01 (July 2002): 187–211. http://dx.doi.org/10.1142/s0219633602000117.

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The sequence and structure of a large body of proteins are becoming increasingly available. It is desirable to explore mathematical tools for efficient extraction of information from such sources. The principles of graph theory, which was earlier applied in fields such as electrical engineering and computer networks are now being adopted to investigate protein structure, folding, stability, function and dynamics. This review deals with a brief account of relevant graphs and graph theoretic concepts. The concepts of protein graph construction are discussed. The manner in which graphs are analyzed and parameters relevant to protein structure are extracted, are explained. The structural and biological information derived from protein structures using these methods is presented.

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Pasupuleti, Murali Krishna. "Graph-Theoretical Intelligence for Adaptive Cybersecurity: Multi-Modal Models, Dynamic Defense, and Federated Threat Detection." International Journal of Academic and Industrial Research Innovations(IJAIRI) 05, no. 04 (April 28, 2025): 347–56. https://doi.org/10.62311/nesx/rp3125.

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Abstract: The rise of sophisticated cyber threats demands adaptive, resilient, and intelligent cybersecurity mechanisms. Graph theory has emerged as a foundational tool for modeling, detecting, and responding to complex attack patterns. This paper critically interprets recent advancements across cybersecurity knowledge graphs, graph neural networks, attack graphs, and federated graph learning. A multi-modal, graph-theoretical intelligence framework is proposed to enhance dynamic defense and collaborative threat detection while maintaining privacy. Statistical analysis, visual models, and new scientific methods are incorporated to demonstrate the feasibility and future potential of graph-based adaptive cybersecurity systems. Keywords: Keywords: Graph Theory, Cybersecurity, Knowledge Graphs, Federated Learning, Attack Graphs, Adaptive Defense, Threat Detection.

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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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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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K, SIVARANJANI, and Shanmuga Sundaram Olappalyam Vaiyapuri. "AN ANALYSIS OF THE SEIDEL LAPLACIAN ENERGY OF A FUZZY INTUITIONISTIC SYSTEM." Suranaree Journal of Science and Technology 32, no. 1 (March 24, 2025): 0010350(1–14). https://doi.org/10.55766/sujst7197.

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This paper presents one of the latest research works in intuitionistic fuzzy graph theory. Along with questionable concepts conveyed in distinctive languages, intuitionistic fuzzy set theory offers a noteworthy and ground-breaking depiction of vulnerability estimation. The concept of energy is related to the spectrum of a graph. The energy of graphs plays a vital role in graph theory. In mathematics, the total sum of the absolute values of the eigenvalues of the graph’s adjacency matrix is referred to as the graph’s energy. In the framework of spectral graph theory, this quantity is extensively studied. When analyzing various repulsive and appealing circumstances, energy graphs are a great tool. Analogous energies based on the eigenvalues of various additional graph matrices have been examined recently. Graph energy possesses numerous mathematical characteristics which are investigated in this paper. One type of graph energy is Seidel Laplacian energy. The Seidel Laplacian energy (SLE) of a graph has already been extended to fuzzy graphs. In this manuscript, it is extended to an intuitionistic fuzzy graph. Energy, Laplacian energy, and SLE of an intuitionistic fuzzy graph are evaluated using some examples, and their lower and upper bounds are derived.

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Tola, Keneni Abera, V. N. Srinivasa Rao Repalle, and Mamo Abebe Ashebo. "Theory and Application of Interval-Valued Neutrosophic Line Graphs." Journal of Mathematics 2024 (March 19, 2024): 1–17. http://dx.doi.org/10.1155/2024/5692756.

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Neutrosophic graphs are used to model inconsistent information and imprecise data about any real-life problem. It is regarded as a generalization of intuitionistic fuzzy graphs. Since interval-valued neutrosophic sets are more accurate, compatible, and flexible than single neutrosophic sets, interval-valued neutrosophic graphs (IVNGs) were defined. The interval-valued neutrosophic graph is a fundamental issue in graph theory that has wide applications in the real world. Also, problems may arise when partial ignorance exists in the datasets of membership [0, 1], and then, the concept of IVNG is crucial to represent the problems. Line graphs of neutrosophic graphs are significant due to their ability to represent and analyze uncertain or indeterminate information about edge relationships and complex networks in graphs. However, there is a research gap on the line graph of interval-valued neutrosophic graphs. In this paper, we introduce the theory of an interval-valued neutrosophic line graph (IVNLG) and its application. In line with that, some mathematical properties such as weak vertex isomorphism, weak edge isomorphism, effective edge, and other properties of IVNLGs are proposed. In addition, we defined the vertex degree of IVNLG with some properties, and by presenting several theorems and propositions, the relationship between fuzzy graph extensions and IVNLGs was explored. Finally, an overview of the algorithm used to solve the problems and the practical application of the introduced graphs were provided.

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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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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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Hammond, David K., Pierre Vandergheynst, and Rémi Gribonval. "Wavelets on graphs via spectral graph theory." Applied and Computational Harmonic Analysis 30, no. 2 (March 2011): 129–50. http://dx.doi.org/10.1016/j.acha.2010.04.005.

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40

Redmon, Eric, Miles Mena, Megan Vesta, Alvi Renzyl Cortes, Lauren Gernes, Simon Merheb, Nick Soto, Chandler Stimpert, and Amanda Harsy. "Optimal Tilings of Bipartite Graphs Using Self-Assembling DNA." PUMP Journal of Undergraduate Research 6 (March 13, 2023): 124–50. http://dx.doi.org/10.46787/pump.v6i0.2427.

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Motivated by the recent advancements in nanotechnology and the discovery of new laboratory techniques using the Watson-Crick complementary properties of DNA strands, formal graph theory has recently become useful in the study of self-assembling DNA complexes. Construction methods based on graph theory have resulted in significantly increased efficiency. We present the results of applying graph theoretical and linear algebra techniques for constructing crossed-prism graphs, crown graphs, book graphs, stacked book graphs, and helm graphs, along with kite, cricket, and moth graphs. In particular, we explore various design strategies for these graph families in two sets of laboratory constraints.

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Vidyasankar, K. "Unified Theory of Database Serializability12." Fundamenta Informaticae 14, no. 2 (February 1, 1991): 147–83. http://dx.doi.org/10.3233/fi-1991-14202.

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A database system is a collection of data items, read or written by transactions in a possibly interleaved fashion. An interleaved execution is assumed to be correct if the sequence of the steps of the transactions, called history, is serializable, that is, the effect of the execution is equivalent to that of some serial execution of the same transactions. In this paper we give a new characterization of serializability that brings out the inherent problem of serialization explicitly. We then give a graph-theoretic analogue of serializable histories. We define a new class of graphs, called serializable graphs, whose properties are such that (i) a serializable graph can be associated with each serializable history, and this can be done for various notions of serializability of histories and for serializability under various sets of constraints, and (ii) a serializable history, in fact a serial one, can be associated with each serializable graph. We use serializable graphs to characterize, in an intuitive manner, serializable histories involving general multi-step transactions, where the same data item can be accessed by several read and write steps of a transaction in an arbitrary manner, and those involving nested transactions. We also define a new notion of serializability for nested transactions. This enables relating several acceptable concurrent executions of transactions, that are not serializable with the traditional transaction concept, to sequential behaviour. Serializability under this notion is also characterized. The main graph-theoretic properties used in these characterizations are a directed cutset matching property and graph contraction.

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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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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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Quinn, Anne. "Using Apps to Visualize Graph Theory." Mathematics Teacher 108, no. 8 (April 2015): 626–31. http://dx.doi.org/10.5951/mathteacher.108.8.0626.

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Jin, Ming, Heng Chang, Wenwu Zhu, and Somayeh Sojoudi. "Power up! Robust Graph Convolutional Network via Graph Powering." Proceedings of the AAAI Conference on Artificial Intelligence 35, no. 9 (May 18, 2021): 8004–12. http://dx.doi.org/10.1609/aaai.v35i9.16976.

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Graph convolutional networks (GCNs) are powerful tools for graph-structured data. However, they have been recently shown to be vulnerable to topological attacks. To enhance adversarial robustness, we go beyond spectral graph theory to robust graph theory. By challenging the classical graph Laplacian, we propose a new convolution operator that is provably robust in the spectral domain and is incorporated in the GCN architecture to improve expressivity and interpretability. By extending the original graph to a sequence of graphs, we also propose a robust training paradigm that encourages transferability across graphs that span a range of spatial and spectral characteristics. The proposed approaches are demonstrated in extensive experiments to simultaneously improve performance in both benign and adversarial situations.

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Kitaev, Sergey, and Haoran Sun. "Human-verifiable proofs in the theory of word-representable graphs." RAIRO - Theoretical Informatics and Applications 58 (2024): 9. http://dx.doi.org/10.1051/ita/2024004.

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A graph is word-representable if it can be represented in a certain way using alternation of letters in words. Word-representable graphs generalise several important and well-studied classes of graphs, and they can be characterised by semi-transitive orientations. Recognising word-representability is an NP-complete problem, and the bottleneck of the theory of word-representable graphs is convincing someone that a graph is non-word-representable, keeping in mind that references to (even publicly available and user-friendly) software are not always welcome. (Word-representability can be justified by providing a semi-transitive orientation as a certificate that can be checked in polynomial time). In the literature, a variety of (usually ad hoc) proofs of non-word-representability for particular graphs, or families of graphs, appear, but for a randomly selected graph, one should expect looking at O(2#edges) orientations and justifying that none of them is semi-transitive. Even if computer would print out all these orientations and would point out what is wrong with each of the orientations, such a proof would be essentially non-checkable by a human. In this paper, we develop methods for an automatic search of human-verifiable proofs of graph non-word-representability. As a proof-of-concept, we provide “short” proofs of non-word-representability, generated automatically by our publicly available user-friendly software, of the Shrikhande graph on 16 vertices and 48 edges (6 “lines” of proof) and the Clebsch graph on 16 vertices and 40 edges (10 “lines” of proof). Producing such short proofs for relatively large graphs would not be possible without the instrumental tool we introduce (allowing to assume orientations of several edges in a graph, not just one edge as it was previously used) that is a game changer in the area. As a bi-product of our studies, we correct two mistakes published multiple times (two graphs out of the 25 non-word-representable graphs on 7 vertices were actually word-representable, while two non-word-representable graphs on 7 vertices were missing).

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MACKIE, IAN, and DETLEF PLUMP. "Theory and applications of term graph rewriting: introduction." Mathematical Structures in Computer Science 17, no. 3 (June 2007): 361–62. http://dx.doi.org/10.1017/s0960129507006081.

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Term graph rewriting is concerned with the representation of functional expressions as graphs and the evaluation of these expressions by rule-based graph transformation. The advantage of computing with graphs rather than terms is that common subexpressions can be shared, improving the efficiency of computations in space and time. Sharing is ubiquitous in implementations of programming languages: many functional, logic, object-oriented and concurrent calculi are implemented using term graphs.

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Akram, Muhammad, Wieslaw A. Dudek, and M. Murtaza Yousaf. "Regularity in Vague Intersection Graphs and Vague Line Graphs." Abstract and Applied Analysis 2014 (2014): 1–10. http://dx.doi.org/10.1155/2014/525389.

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Fuzzy graph theory is commonly used in computer science applications, particularly in database theory, data mining, neural networks, expert systems, cluster analysis, control theory, and image capturing. A vague graph is a generalized structure of a fuzzy graph that gives more precision, flexibility, and compatibility to a system when compared with systems that are designed using fuzzy graphs. In this paper, we introduce the notion of vague line graphs, and certain types of vague line graphs and present some of their properties. We also discuss an example application of vague digraphs.

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M., Vasuki, and Dinesh Kumar A. "GRAPH THEORY WITH THEIR REAL-TIME APPLICATIONS IN EVERYDAY LIFE AND TECHNOLOGY." International Journal of Engineering Research and Modern Education (IJERME) 6, no. 1 (June 8, 2021): 17–20. https://doi.org/10.5281/zenodo.6079366.

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A petal graph is a connected graph G with maximum degree three, minimum degree two, and such that the set of vertices of degree three induces a 2–regular graph and the whole set of vertices of degree in two induces an empty graph. We prove here that, with the single exception of the graph obtained from the Petersen graph by deleting one vertex, all petal graphs are Class 1.

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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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<abstract><p>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.</p></abstract>

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PUCANOVIĆ, ZORAN S., MARKO RADOVANOVIĆ, and ALEKSANDRA LJ ERIĆ. "ON THE GENUS OF THE INTERSECTION GRAPH OF IDEALS OF A COMMUTATIVE RING." Journal of Algebra and Its Applications 13, no. 05 (February 25, 2014): 1350155. http://dx.doi.org/10.1142/s0219498813501557.

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To each commutative ring R one can associate the graph G(R), called the intersection graph of ideals, whose vertices are nontrivial ideals of R. In this paper, we try to establish some connections between commutative ring theory and graph theory, by study of the genus of the intersection graph of ideals. We classify all graphs of genus 2 that are intersection graphs of ideals of some commutative rings and obtain some lower bounds for the genus of the intersection graph of ideals of a nonlocal commutative ring.

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