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Vertex (graph theory)

About: Vertex (graph theory) is a research topic. Over the lifetime, 17595 publications have been published within this topic receiving 255973 citations. The topic is also known as: node & graph vertex.


Papers
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Book
03 Jul 2003
TL;DR: This chapter discusses probabilistic ingredients, the largest component for a binomial process, and connectedivity and the number of components in a graph-like model.
Abstract: 1. Introduction 2. Probabilistic ingredients 3. Subgraph and component counts 4. Typical vertex degrees 5. Geometrical ingredients 6. Maximum degree, cliques and colourings 7. Minimum degree: laws of large numbers 8. Minimum degree: convergence in distribution 9. Percolative ingredients 10. Percolation and the largest component 11. The largest component for a binomial process 12. Ordering and partitioning problems 13. Connectivity and the number of components References Index

2,271 citations

Book
23 Jun 1989
TL;DR: In this paper, a connected simple graph with vertex set X of diameter d is considered, and the authors define Ri X2 by (x, y) Ri whenever x and y have graph distance.
Abstract: Consider a connected simple graph with vertex set X of diameter d. Define Ri X2 by (x, y) Ri whenever x and y have graph distance

2,264 citations

Journal ArticleDOI
TL;DR: The method yields polynomial algorithms for vertex packing in perfect graphs, for the matching and matroid intersection problems, for optimum covering of directed cuts of a digraph, and for the minimum value of a submodular set function.
Abstract: L. G. Khachiyan recently published a polynomial algorithm to check feasibility of a system of linear inequalities. The method is an adaptation of an algorithm proposed by Shor for non-linear optimization problems. In this paper we show that the method also yields interesting results in combinatorial optimization. Thus it yields polynomial algorithms for vertex packing in perfect graphs; for the matching and matroid intersection problems; for optimum covering of directed cuts of a digraph; for the minimum value of a submodular set function; and for other important combinatorial problems. On the negative side, it yields a proof that weighted fractional chromatic number is NP-hard.

2,170 citations

Journal Article
TL;DR: In this paper, complex realizations of vertex operator algebraic expressions are presented, and the main theorem of complex realisation of vertices operator algebra is proved. But the complexity is not discussed.
Abstract: Lie Algebras. Formal Calculus: Introduction. Realizations of sl(2) by Twisted Vertex Operators. Realizations of sl(2) by Untwisted Vertex Operators. Central Extensions. The Simple Lie Algebras An, Dn, En. Vertex Operator Realizations of An, Dn, En. General Theory of Untwisted Vertex Operators. General Theory of Twisted Vertex Operators. The Moonshine Module. Triality. The Main Theorem. Completion of the Proof. Appendix: Complex Realization of Vertex Operator Algebras. Bibliography. Index of frequently used symbols. Index.

2,010 citations

Book
01 Jan 1987
TL;DR: The Strong Perfect Graph Conjecture as discussed by the authors is based on the strong perfect graph conjecture, which is a generalization of the concept of generalized perfection, generalized perfection and related concepts.
Abstract: Preface 1. Basic Concepts 2. Perfection, Generalized Perfection, and Related Concepts 3. Cycles, Chords and Bridges 4. Models and Interactions 5. Vertex and Edge Orderings 6. Posets 7. Forbidden Subgraphs 8. Hypergraphs and Graphs 9. Matrices and Polyhedra 10. Distance Properties 11. Algebraic Compositions and Recursive Definitions 12. Decompositions and Cutsets 13. Threshold Graphs and Related Concepts 14. The Strong Perfect Graph Conjecture Appendix A. Recognition Appendix B. Containment Relationships Bibliography Index.

1,958 citations


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Performance
Metrics
No. of papers in the topic in previous years
YearPapers
20242
20231,918
20224,076
20211,489
20201,310
20191,188