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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
From a Zoo to a Zoology: Towards a General Theory of Graph Polynomials
TL;DR: Two classes of (hyper)graph polynomials definable in second order logic are introduced, and a research program for their classification in terms of definability and complexity considerations, and various notions of reducibilities are outlined.
Journal ArticleDOI
A Decomposition Theory for Binary Linear Codes
TL;DR: An overview of this code decomposition theory of matroids is given, and some of its implications in the context of the recently discovered formulation of maximum-likelihood (ML) decoding of a binary linear code over a binary-input discrete memoryless channel as a linear programming problem is discussed.
Proceedings ArticleDOI
Multicast time maximization in energy constrained wireless networks
TL;DR: It is shown that lifetime-maximizing static power assignments can be found in polynomial time, whereas for dynamic assignments, a quantized-time version of the problem is NP-hard.
Journal ArticleDOI
The role of nonlinearity in computing graph-theoretical properties of resting-state functional magnetic resonance imaging brain networks
TL;DR: A comparison of network analysis results obtained from the brain connectivity graphs capturing either full (both linear and nonlinear) or only linear connectivity using 24 sessions of human resting-state fMRI shows that the nonlinearity effect is practically negligible when compared to the intersubject variability of the graph measures.
Journal ArticleDOI
The semigroup approach to transport processes in networks
TL;DR: In this paper, the operator semigroups were used to study transport processes in networks and applied to a linear Boltzmann equation on a finite as well as on an infinite network and yields wellposedness and information on the long term behavior of the solutions to the presented problems.