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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
Citations
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Journal ArticleDOI
Tight upper tail bounds for cliques
Bobby DeMarco,Jeff Kahn +1 more
TL;DR: With ξk =ξ kn,p the number of copies of Kk in the usual (Erdős‐Rényi) random graph G(n,p), p ≥ n‐2/(k‐1) and η > 0, this is tight up to the value of the constant in the exponent.
Proceedings ArticleDOI
Span programs for functions with constant-sized 1-certificates: extended abstract
TL;DR: In this article, the authors proposed a quantum algorithm for the triangle problem with query complexity O(n35/27) that is better than O( n13/10) of the best known algorithm by Magniez et al.
Book ChapterDOI
Random Graphs and Branching Processes
Béla Bollobás,Oliver Riordan +1 more
TL;DR: A host of inhomogeneous random graph models have been constructed and studied in order to model large-scale real-world networks such as the world wide web, neural networks, and social networks.
Proceedings ArticleDOI
On the stability and optimality of distributed Kalman filters with finite-time data fusion
Usman A. Khan,Ali Jadbabaie +1 more
TL;DR: In this article, the authors consider distributed estimation for discrete-time, linear systems, with finite-time data fusion of agent measurements between each time-step of the dynamics, and show that the observation map at each agent is a linear combination of the local observation maps, and that this new observation map is observable (if the data is fused for a sufficient number of iterations that we lower bound).
Proceedings ArticleDOI
On a dynamic extension of the theory of graphs
TL;DR: In this article, the authors propose a framework for studying dynamic graphs as an area that lies at the intersection of dynamical systems and combinatorics, and highlight the directions which are particularly promising in this venue.