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Journal articles on the topic 'Graph theory'

Author: Grafiati

Published: 4 June 2021

Last updated: 12 October 2024

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1

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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2

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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3

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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4

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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5

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

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9

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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12

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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14

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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15

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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17

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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18

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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19

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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20

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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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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22

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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23

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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24

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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25

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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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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27

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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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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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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31

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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33

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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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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35

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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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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Chen, Lin, Li Zeng, Jin Peng, Junren Ming, and Xianghui Zhu. "Regularity Index of Uncertain Random Graph." Symmetry 15, no. 1 (January 3, 2023): 137. http://dx.doi.org/10.3390/sym15010137.

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A graph containing some edges with probability measures and other edges with uncertain measures is referred to as an uncertain random graph. Numerous real-world problems in social networks and transportation networks can be boiled down to optimization problems in uncertain random graphs. Actually, information in optimization problems in uncertain random graphs is always asymmetric. Regularization is a common optimization problem in graph theory, and the regularity index is a fundamentally measurable indicator of graphs. Therefore, this paper investigates the regularity index of an uncertain random graph within the framework of chance theory and information asymmetry theory. The concepts of k-regularity index and regularity index of the uncertain random graph are first presented on the basis of the chance theory. Then, in order to compute the k-regularity index and the regularity index of the uncertain random graph, a simple and straightforward calculating approach is presented and discussed. Furthermore, we discuss the relationship between the regularity index and the k-regularity index of the uncertain random graph. Additionally, an adjacency matrix-based algorithm that can compute the k-regularity index of the uncertain random graph is provided. Some specific examples are given to illustrate the proposed method and algorithm. Finally, we conclude by highlighting some potential applications of uncertain random graphs in social networks and transportation networks, as well as the future vision of its combination with symmetry.

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38

Hussain, Muhammad Tanveer, Navid Iqbal, Usman Babar, Tabasam Rashid, and Md Nur Alam. "On Sum Degree-Based Topological Indices of Some Connected Graphs." Mathematical Problems in Engineering 2022 (July 30, 2022): 1–19. http://dx.doi.org/10.1155/2022/2961173.

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In mathematical chemistry, molecular structure of any chemical substance can be expressed by a numeric number or polynomial or sequence of number which represents the whole graph is called topological index. An important branch of graph theory is the chemical graph theory. As a consequence of their worldwide uses, chemical networks have inspired researchers since their development. Determination of the expressions for topological indices of different derived graphs of graphs is a new and interesting problem in graph theory. In this article, some graphs which are derived from honeycomb structure are studied and obtained their exact results for sum degree-based indices. Additionally, a comparison is shown graphically among all the indices.

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Alali, Amal S., Shahbaz Ali, Noor Hassan, Ali M. Mahnashi, Yilun Shang, and Abdullah Assiry. "Algebraic Structure Graphs over the Commutative Ring Zm: Exploring Topological Indices and Entropies Using M-Polynomials." Mathematics 11, no. 18 (September 7, 2023): 3833. http://dx.doi.org/10.3390/math11183833.

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The field of mathematics that studies the relationship between algebraic structures and graphs is known as algebraic graph theory. It incorporates concepts from graph theory, which examines the characteristics and topology of graphs, with those from abstract algebra, which deals with algebraic structures such as groups, rings, and fields. If the vertex set of a graph G^ is fully made up of the zero divisors of the modular ring Zn, the graph is said to be a zero-divisor graph. If the products of two vertices are equal to zero under (modn), they are regarded as neighbors. Entropy, a notion taken from information theory and used in graph theory, measures the degree of uncertainty or unpredictability associated with a graph or its constituent elements. Entropy measurements may be used to calculate the structural complexity and information complexity of graphs. The first, second and second modified Zagrebs, general and inverse general Randics, third and fifth symmetric divisions, harmonic and inverse sum indices, and forgotten topological indices are a few topological indices that are examined in this article for particular families of zero-divisor graphs. A numerical and graphical comparison of computed topological indices over a proposed structure has been studied. Furthermore, different kinds of entropies, such as the first, second, and third redefined Zagreb, are also investigated for a number of families of zero-divisor graphs.

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Huang, Liangsong, Yu Hu, Yuxia Li, P. K. Kishore Kumar, Dipak Koley, and Arindam Dey. "A Study of Regular and Irregular Neutrosophic Graphs with Real Life Applications." Mathematics 7, no. 6 (June 17, 2019): 551. http://dx.doi.org/10.3390/math7060551.

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Fuzzy graph theory is a useful and well-known tool to model and solve many real-life optimization problems. Since real-life problems are often uncertain due to inconsistent and indeterminate information, it is very hard for an expert to model those problems using a fuzzy graph. A neutrosophic graph can deal with the uncertainty associated with the inconsistent and indeterminate information of any real-world problem, where fuzzy graphs may fail to reveal satisfactory results. The concepts of the regularity and degree of a node play a significant role in both the theory and application of graph theory in the neutrosophic environment. In this work, we describe the utility of the regular neutrosophic graph and bipartite neutrosophic graph to model an assignment problem, a road transport network, and a social network. For this purpose, we introduce the definitions of the regular neutrosophic graph, star neutrosophic graph, regular complete neutrosophic graph, complete bipartite neutrosophic graph, and regular strong neutrosophic graph. We define the d m - and t d m -degrees of a node in a regular neutrosophic graph. Depending on the degree of the node, this paper classifies the regularity of a neutrosophic graph into three types, namely d m -regular, t d m -regular, and m-highly irregular neutrosophic graphs. We present some theorems and properties of those regular neutrosophic graphs. The concept of an m-highly irregular neutrosophic graph on cycle and path graphs is also investigated in this paper. The definition of busy and free nodes in a regular neutrosophic graph is presented here. We introduce the idea of the μ -complement and h-morphism of a regular neutrosophic graph. Some properties of complement and isomorphic regular neutrosophic graphs are presented here.

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Sesa, Junianto, and Siswanto Siswanto. "Determination of Fractional Chromatic Numbers in the Operation of Adding Two Different Graphs." Jurnal Matematika, Statistika dan Komputasi 18, no. 2 (January 1, 2022): 161–68. http://dx.doi.org/10.20956/j.v18i2.14501.

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The development of graph theory has provided many new pieces of knowledge, one of them is graph color. Where the application is spread in various fields such as the coding index theory. Fractional coloring is multiple coloring at points with different colors where the adjoining point has a different color. The operation in the graph is known as the sum operation. Point coloring can be applied to graphs where the result of operations is from several special graphs. In this case, the graph summation results of the path graph and the cycle graph will produce the same fractional chromatic number as the sum of the fractional chromatic numbers of each graph before it is operated.

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Harary, Frank, and Hong-Sen Yan. "Logical Foundations of Kinematic Chains: Graphs, Line Graphs, and Hypergraphs." Journal of Mechanical Design 112, no. 1 (March 1, 1990): 79–83. http://dx.doi.org/10.1115/1.2912583.

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In terms of concepts from the theory of graphs and hypergraphs we formulate a precise structural characterization of a kinematic chain. To do this, we require the operations of line graph, intersection graph, and hypergraph duality. Using these we develop simple algorithms for constructing the unique graph G (KC) of a kinematic chain KC and (given an admissible graph G) for forming the unique kinematic chain whose graph is G. This one-to-one correspondence between kinematic chains and a class of graphs enables the mathematical and logical power, precision, concepts, and theorems of graph theory to be applied to gain new insights into the structure of kinematic chains.

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Snehal Eknath Phule. "Graph Theory Applications in Database Management." International Journal of Scientific Research in Modern Science and Technology 3, no. 3 (March 16, 2024): 13–17. http://dx.doi.org/10.59828/ijsrmst.v3i3.190.

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Graph theory, which is a branch of discrete mathematics, has emerged as a powerful tool in various domains, including database management. This abstract investigates the ways in which ideas and methods from graph theory which can be applied to database systems, offering a thorough synopsis of their benefits. Complex interactions within data can be well-modeled by using the basic concepts of graph theory, such as nodes, edges, and relationships. Because of its capacity to represent and query complex relationships, graph databases have become more and more popular in the field of database administration. Graph databases are well-suited for situations such as social networks, recommendation systems, and interconnected data domains because they are excellent at representing and traversing relationships, in contrast to standard relational databases, which are excellent at managing structured data. The abstract delves into the key graph-based data models, such as property graphs, RDF (Resource Description Framework), explaining how they facilitate the representation of diverse relationships. Furthermore, it explores the efficient storage and retrieval mechanisms that leverage graph traversal algorithms to extract valuable insights from interconnected datasets. The document highlights specific use cases where graph theory contributes to database management, including fraud detection, social network analysis, and recommendation systems. Additionally, it discusses the challenges associated with integrating graph databases into existing infrastructures and proposes solutions to address scalability and performance concerns. The abstract also touches upon the advancements in graph database query languages (Cypher) and SPARQL, showcasing their expressive power in querying complex relationships. The inclusion of graph-based indexing and optimization techniques demonstrates how database systems can efficiently handle queries involving large-scale graph data. As graph databases continue to evolve, this abstract concludes by outlining potential future directions in the intersection of graph theory and database management. It emphasizes the importance of ongoing research in developing scalable and efficient solutions for managing interconnected data, ultimately paving the way for more sophisticated and context-aware database systems relationships. Furthermore, it explores the efficient storage and retrieval mechanisms that leverage graph traversal algorithms to extract valuable insights from interconnected datasets.

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Wang, Xingxing, Mengjie Li, Wenxue Shi, Yue Wu, and Yuting Zhao. "Improvement of Sufficient Conditions for Pancyclic Graphs." Scientific Journal of Technology 6, no. 1 (January 22, 2024): 29–32. http://dx.doi.org/10.54691/w10e3408.

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The study of cyclic graphs has always been a hot topic in the field of graph theory and has received widespread attention from graph theory practitioners. If an n-order graph G exactly contains cycles of all lengths from 3 to n, then the graph is called a pan cycle graph. This article proves that, after excluding some special cases, when the number of edges in graph G is greater than or equal to C2n-3+12, graph G must be a pan cyclic graph.

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Hosamani, S. M., V. B. Awati, and R. M. Honmore. "On graphs with equal dominating and c-dominating energy." Applied Mathematics and Nonlinear Sciences 4, no. 2 (December 24, 2019): 503–12. http://dx.doi.org/10.2478/amns.2019.2.00047.

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AbstractGraph energy and domination in graphs are most studied areas of graph theory. In this paper we try to connect these two areas of graph theory by introducing c-dominating energy of a graph G. First, we show the chemical applications of c-dominating energy with the help of well known statistical tools. Next, we obtain mathematical properties of c-dominating energy. Finally, we characterize trees, unicyclic graphs, cubic and block graphs with equal dominating and c-dominating energy.

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Thenge, J. D., B. Surendranath Reddy, and Rupali S. Jain. "Contribution to Soft Graph and Soft Tree." New Mathematics and Natural Computation 15, no. 01 (December 25, 2018): 129–43. http://dx.doi.org/10.1142/s179300571950008x.

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Soft set theory introduced by D. Molodstov is a new theory which deals with uncertainty. Connected graphs can be represented by using soft sets called soft graphs. In the present paper, we introduce the tabular representation of soft graph and define radius, diameter, center and degree of soft graph. We also define union, product of soft graphs and soft trees. We then derive some properties of radius, degree of vertex in soft graph and soft trees.

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Catlin, Paul A., and Gary Chartrand. "Introductory Graph Theory." American Mathematical Monthly 94, no. 5 (May 1987): 483. http://dx.doi.org/10.2307/2322751.

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Lloyd, E. Keith, and L. R. Foulds. "Graph Theory Applications." Mathematical Gazette 78, no. 481 (March 1994): 95. http://dx.doi.org/10.2307/3619469.

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Diaconis, Persi, and Bela Bollobas. "Modern Graph Theory." Journal of the American Statistical Association 95, no. 452 (December 2000): 1377. http://dx.doi.org/10.2307/2669801.

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Wilson, B. J. "ALGORITHMIC GRAPH THEORY." Bulletin of the London Mathematical Society 18, no. 6 (November 1986): 630–31. http://dx.doi.org/10.1112/blms/18.6.630.

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