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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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Proceedings ArticleDOI
Hamiltonian-like Properties of k-Ary n-Cubes
TL;DR: It is proved that k-ary 2-cube is almost Hamiltonian- connected, bipanconnected, bipancyclic and Hamiltonianconnected if k is even and k is odd.
Posted Content
Three-variable expanding polynomials and higher-dimensional distinct distances
TL;DR: In this article, it was shown that for a quadratic polynomial, which is not of the form g(h(x)+k(y)+l(z)), there is a lower bound of Ω(n 3/2 ) on the distance required to obtain a Cartesian product that determines almost $|A|^2$ distinct distances if the characteristic of the field is not too large.
Journal ArticleDOI
Memory effects in schematic models of glasses subjected to oscillatory deformation
TL;DR: Two schematic models of glasses subjected to oscillatory shear deformation are considered, motivated by the observations in computer simulations of a model glass, of a nonequilibrium transition from a localized to a diffusive regime as the shear amplitude is increased, and of persistent memory effects in the localized regime.
Journal ArticleDOI
Worst-Case Diagnosis Completeness in Regular Graphs under the PMC Model
Antonio Caruso,Stefano Chessa +1 more
TL;DR: A lower bound to the worst-case diagnosis completeness for regular graphs for which vertex- isoperimetric inequalities are known is provided and it is shown how this bound can be applied to toroidal grids.
Dissertation
Approximate inference in gaussian graphical models
Alan S. Willsky,Dmitry Malioutov +1 more
TL;DR: A walk-sum interpretation is developed for a popular distributed approximate inference algorithm called loopy belief propagation (LBP), and conditions for its convergence are established to analyze more powerful versions of LBP that trade off convergence and accuracy for computational complexity, and establish conditions for their convergence.