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Modern graph theory
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This book presents an account of newer topics, including Szemer'edi's Regularity Lemma and its use; Shelah's extension of the Hales-Jewett Theorem; the precise nature of the phase transition in a random graph process; the connection between electrical networks and random walks on graphs; and the Tutte polynomial and its cousins in knot theory.Abstract:
The time has now come when graph theory should be part of the education of every serious student of mathematics and computer science, both for its own sake and to enhance the appreciation of mathematics as a whole. This book is an in-depth account of graph theory, written with such a student in mind; it reflects the current state of the subject and emphasizes connections with other branches of pure mathematics. The volume grew out of the author's earlier book, Graph Theory -- An Introductory Course, but its length is well over twice that of its predecessor, allowing it to reveal many exciting new developments in the subject. Recognizing that graph theory is one of several courses competing for the attention of a student, the book contains extensive descriptive passages designed to convey the flavor of the subject and to arouse interest. In addition to a modern treatment of the classical areas of graph theory such as coloring, matching, extremal theory, and algebraic graph theory, the book presents a detailed account of newer topics, including Szemer\'edi's Regularity Lemma and its use, Shelah's extension of the Hales-Jewett Theorem, the precise nature of the phase transition in a random graph process, the connection between electrical networks and random walks on graphs, and the Tutte polynomial and its cousins in knot theory. In no other branch of mathematics is it as vital to tackle and solve challenging exercises in order to master the subject. To this end, the book contains an unusually large number of well thought-out exercises: over 600 in total. Although some are straightforward, most of them are substantial, and others will stretch even the most able reader.read more
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Journal ArticleDOI
Enumerative Properties of Ferrers Graphs
TL;DR: In this paper, the authors define a class of bipartite graphs that correspond naturally with Ferrers diagrams and give expressions for the number of spanning trees, the Hamiltonian paths when applicable, the chromatic polynomial and chromatic symmetric function.
Journal ArticleDOI
Moduli of McKay quiver representations II: Gröbner basis techniques
TL;DR: In this article, the moduli spaces of the McKay quiver representations were studied using Grobner bases and toric geometry, and a simple description of the quiver representation corresponding to the torus orbits of Y θ was given.
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Structural and topological phase transitions on the German Stock Exchange
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Fault-Tolerant Geometric Spanners
Artur Czumaj,Hairong Zhao +1 more
TL;DR: A greedy algorithm is presented that for any t > 1 and any non-negative integer k, constructs a k-fault-tolerant t-spanner in which every vertex is of degree O(k) and whose total cost is O( k2) times the cost of the minimum spanning tree; these bounds are asymptotically optimal.
Journal ArticleDOI
Identifying effective multiple spreaders by coloring complex networks
TL;DR: This Letter makes an attempt to find effective multiple spreaders in complex networks by generalizing the idea of the coloring problem in graph theory to complex networks, and finds that the method is more effective in accelerating the spreading process and maximizing the spreading coverage than the traditional method.