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INTRODUCTION

Mathew Penrose

in Random Geometric Graphs

Published in print:
2003
Published Online:
September 2007
ISBN:
9780198506263
eISBN:
9780191707858
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780198506263.003.0001
Subject:
Mathematics, Probability / Statistics

This introductory chapter contains a general discussion of both the historical and the applied background behind the study of random geometric graphs. A brief overview is presented, along with some ... More


FURTHER ADVANCED TECHNIQUES

Rolf Niedermeier

in Invitation to Fixed-Parameter Algorithms

Published in print:
2006
Published Online:
September 2007
ISBN:
9780198566076
eISBN:
9780191713910
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780198566076.003.0011
Subject:
Mathematics, Combinatorics / Graph Theory / Discrete Mathematics

This chapter surveys further important techniques for designing fixed-parameter algorithms. These include color-coding, integer linear programming, iterative compression, greedy localization, and ... More


FOURIER ANALYSIS ON FINITE ABELIAN GROUPS: SOME GRAPHICAL APPLICATIONS

Andrew Goodall

in Combinatorics, Complexity, and Chance: A Tribute to Dominic Welsh

Published in print:
2007
Published Online:
September 2007
ISBN:
9780198571278
eISBN:
9780191718885
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780198571278.003.0007
Subject:
Mathematics, Probability / Statistics

This article reviews basic techniques of Fourier analysis on a finite abelian group Q, with subsequent applications in graph theory. These include evaluations of the Tutte polynomial of a graph G in ... More


Graph Theory: A Look Back—The Road Ahead

Arthur Benjamin, Gary Chartrand, and Ping Zhang

in The Fascinating World of Graph Theory

Published in print:
2017
Published Online:
May 2018
ISBN:
9780691175638
eISBN:
9781400852000
Item type:
chapter
Publisher:
Princeton University Press
DOI:
10.23943/princeton/9780691175638.003.0013
Subject:
Mathematics, Applied Mathematics

This book concludes with an epilogue, which traces the evolution of graph theory, from the conceptualization of the Königsberg Bridge Problem and its generalization by Leonhard Euler, whose solution ... More


Algorithmic aspects

J. L. Ramírez Alfonsín

in The Diophantine Frobenius Problem

Published in print:
2005
Published Online:
September 2007
ISBN:
9780198568209
eISBN:
9780191718229
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780198568209.003.0001
Subject:
Mathematics, Algebra, Combinatorics / Graph Theory / Discrete Mathematics

This chapter is devoted to the computational aspects of the Frobenius number. After discussing a number of methods to solve FP when n = 3 (some of these procedures make use of diverse concepts, such ... More


The Fascinating World of Graph Theory

Ping Zhang, Gary Chartrand, and Arthur Benjamin

Published in print:
2017
Published Online:
May 2018
ISBN:
9780691175638
eISBN:
9781400852000
Item type:
book
Publisher:
Princeton University Press
DOI:
10.23943/princeton/9780691175638.001.0001
Subject:
Mathematics, Applied Mathematics

Graph theory goes back several centuries and revolves around the study of graphs—mathematical structures showing relations between objects. With applications in biology, computer science, ... More


Modern graph theory

LOWELL BEINEKE and ROBIN WILSON

in Combinatorics: Ancient and Modern

Published in print:
2013
Published Online:
September 2013
ISBN:
9780199656592
eISBN:
9780191748059
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780199656592.003.0015
Subject:
Mathematics, Combinatorics / Graph Theory / Discrete Mathematics, History of Mathematics

During the first half of the 20th century many classic theorems about graphs were discovered, but it was not until the second half of the century that graph theory emerged as an important field in ... More


Introducing Graphs

Arthur Benjamin, Gary Chartrand, and Ping Zhang

in The Fascinating World of Graph Theory

Published in print:
2017
Published Online:
May 2018
ISBN:
9780691175638
eISBN:
9781400852000
Item type:
chapter
Publisher:
Princeton University Press
DOI:
10.23943/princeton/9780691175638.003.0001
Subject:
Mathematics, Applied Mathematics

This chapter provides an introduction to graphs, a mathematical structure for visualizing, analyzing, and generalizing a situation or problem. It first consider four problems that have a distinct ... More


The Mathematics of Various Entertaining Subjects: Research in Games, Graphs, Counting, and Complexity, Volume 2

Jason Rosenhouse and Jennifer Beineke (eds)

Published in print:
2017
Published Online:
May 2018
ISBN:
9780691171920
eISBN:
9781400889136
Item type:
book
Publisher:
Princeton University Press
DOI:
10.23943/princeton/9780691171920.001.0001
Subject:
Mathematics, History of Mathematics

The history of mathematics is filled with major breakthroughs resulting from solutions to recreational problems. Problems of interest to gamblers led to the modern theory of probability, for example, ... More


Synchronizing Graphs

Arthur Benjamin, Gary Chartrand, and Ping Zhang

in The Fascinating World of Graph Theory

Published in print:
2017
Published Online:
May 2018
ISBN:
9780691175638
eISBN:
9781400852000
Item type:
chapter
Publisher:
Princeton University Press
DOI:
10.23943/princeton/9780691175638.003.0012
Subject:
Mathematics, Applied Mathematics

This chapter considers a new type of graph coloring known as edge coloring. It begins with a discussion of an idea by Scottish physicist Peter Guthrie Tait that led to edge coloring. Tait proved that ... More


Early graph theory

ROBIN WILSON

in Combinatorics: Ancient and Modern

Published in print:
2013
Published Online:
September 2013
ISBN:
9780199656592
eISBN:
9780191748059
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780199656592.003.0009
Subject:
Mathematics, Combinatorics / Graph Theory / Discrete Mathematics, History of Mathematics

The origins of graph theory are humble, even frivolous. Whereas many branches of mathematics were motivated by fundamental problems of calculation, motion, and measurement, the problems which led to ... More


Similarity, graphs and networks, random matrices, and SVD

Peter Grindrod CBE

in Mathematical Underpinnings of Analytics: Theory and Applications

Published in print:
2014
Published Online:
March 2015
ISBN:
9780198725091
eISBN:
9780191792526
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780198725091.003.0002
Subject:
Mathematics, Analysis, Probability / Statistics

This chapter considers how a range of ideas from linear algebra, nonnegative matrices, random matrices and modern graph theory may assist in the understanding of observed pairwise similarities and ... More


Orienting Graphs

Arthur Benjamin, Gary Chartrand, and Ping Zhang

in The Fascinating World of Graph Theory

Published in print:
2017
Published Online:
May 2018
ISBN:
9780691175638
eISBN:
9781400852000
Item type:
chapter
Publisher:
Princeton University Press
DOI:
10.23943/princeton/9780691175638.003.0009
Subject:
Mathematics, Applied Mathematics

This chapter focuses on oriented graphs and their use to represent a sports tournament where assigning a direction to an edge represents the defeat of one team by another. It first considers the 1941 ... More


Combinatorics: Ancient and Modern

Robin Wilson and John J. Watkins (eds)

Published in print:
2013
Published Online:
September 2013
ISBN:
9780199656592
eISBN:
9780191748059
Item type:
book
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780199656592.001.0001
Subject:
Mathematics, Combinatorics / Graph Theory / Discrete Mathematics, History of Mathematics

The history of mathematics is a well-studied and vibrant area of research, with books and scholarly articles published on various aspects of the subject. Yet, the history of combinatorics seems to ... More


Computational Constraints on Modeling in Systems Biology

Vipul Periwal

in System Modeling in Cellular Biology: From Concepts to Nuts and Bolts

Published in print:
2006
Published Online:
August 2013
ISBN:
9780262195485
eISBN:
9780262257060
Item type:
chapter
Publisher:
The MIT Press
DOI:
10.7551/mitpress/9780262195485.003.0015
Subject:
Mathematics, Mathematical Biology

This chapter reviews various computational algorithms and data structures that are relevant for systems biology. It also provides a summary of the computational aspects of sorting, searching, dynamic ... More


Making Walks Count: From Silent Circles to Hamiltonian Cycles

Max A. Alekseyev and Gérard P. Michon

in The Mathematics of Various Entertaining Subjects: Research in Games, Graphs, Counting, and Complexity, Volume 2

Published in print:
2017
Published Online:
May 2018
ISBN:
9780691171920
eISBN:
9781400889136
Item type:
chapter
Publisher:
Princeton University Press
DOI:
10.23943/princeton/9780691171920.003.0010
Subject:
Mathematics, History of Mathematics

Leonhard Euler (1707–1783) invented graph theory in 1735, by solving a puzzle of interest to the inhabitants of Königsberg. The city comprised three distinct land masses, connected by seven bridges. ... More


Introduction

I. S. Duff, A. M. Erisman, and J. K. Reid

in Direct Methods for Sparse Matrices

Published in print:
2017
Published Online:
April 2017
ISBN:
9780198508380
eISBN:
9780191746420
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780198508380.003.0001
Subject:
Mathematics, Numerical Analysis

The use of graph theory to ‘visualize’ the relationship between sparsity patterns and Gaussian elimination is introduced. The potential of significant savings from the exploitation of sparsity is ... More


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