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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
Counting cycles and finite dimensional Lp norms
TL;DR: It is proved that the complete graph is optimal in more ways than could be imagined and sharp bounds for the number of n-cycles in a finite graph are obtained.
Proceedings ArticleDOI
Reciprocity and Barter in Peer-to-Peer Systems
TL;DR: It is proved that any indirect reciprocity schedule of exchanges, in the absence of relays, can be replaced by a direct reciprocity Schedule, provided that users are willing to download undemanded content for bartering purposes and use up to twice the bandwidth they would use under indirect reciprocities.
Journal ArticleDOI
The Deformed Graph Laplacian and Its Applications to Network Centrality Analysis
TL;DR: A new network centrality measure based on the concept of nonbacktracking walks, that is, walks not containing subsequences of the form uvu where u and v are any distinct connected vertices of the underlying graph, is introduced and studied.
Journal ArticleDOI
Topological Graph Polynomial and Quantum Field Theory Part II: Mehler Kernel Theories
TL;DR: In this article, the authors define a new topological polynomial extending the Bollobas-Riordan one, which obeys a four-term reduction relation of the deletion/contraction type and has a natural behaviour under partial duality.
Journal ArticleDOI
Some remarks on the compressed zero-divisor graph
TL;DR: In this article, the authors studied the equivalence relations on R and their corresponding graph-theoretic translations to the compressed zero-divisor graph Γ E (R ).