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EXPANDING THE TUTTE POLYNOMIAL OF A MATROID OVER THE INDEPENDENT SETS

Koko Kalambay Kayibi

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/9780198571278.003.0011
Subject:
Mathematics, Probability / Statistics

This chapter provides direct combinatorial proof of an expansion of the Tutte polynomial by independent sets of the matroid. Another expansion of the Tutte polynomial is presented in terms of ... More


ORBIT COUNTING AND THE TUTTE POLYNOMIAL

Peter J. Cameron

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/9780198571278.003.0001
Subject:
Mathematics, Probability / Statistics

This chapter summarizes the various attempts to extend the Tutte polynomial of a matroid to a polynomial which counts orbits of a group on various sets of objects that the usual Tutte polynomial ... More


TUTTE-WHITNEY POLYNOMIALS: SOME HISTORY AND GENERALIZATIONS

Graham E. Farr

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/9780198571278.003.0003
Subject:
Mathematics, Probability / Statistics

The Tutte-Whitney polynomials play a key role in the study of counting problems on graphs, and have close connections with statistical mechanics and knot theory. This chapter briefly reviews their ... 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


FLOWS AND FERROMAGNETS

Geoffrey Grimmett

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/9780198571278.003.0008
Subject:
Mathematics, Probability / Statistics

The Tutte polynomial and its relatives play important roles in matroid theory, computational complexity, and models of statistical physics. They provide the natural way to count and relate a variety ... More


APPROXIMATING THE TUTTE POLYNOMIAL

Mark Jerrum

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/9780198571278.003.0009
Subject:
Mathematics, Probability / Statistics

This chapter examines some algorithmic problems associated with matroids. It focuses on determining a ‘fully polynomial randomized approximation scheme’ or ‘FPRAS’. First, the problem of counting ... More


COMPLEXITY OF GRAPH POLYNOMIALS

Steven D. Noble

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/9780198571278.003.0013
Subject:
Mathematics, Probability / Statistics

This chapter examines the complexity of evaluating graph polynomials, related to the Tutte polynomial, for various classes of matroids. It begins with a short introduction to matroids, complexity, ... More


THE CONTRIBUTIONS OF DOMINIC WELSH TO MATROID THEORY

James Oxley

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/9780198571278.003.0015
Subject:
Mathematics, Probability / Statistics

Dominic Welsh began writing papers in matroid theory nearly forty years ago. Since then, he has made numerous important contributions to the subject. This chapter reviews Dominic Welsh's work in and ... 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


Higher Connectivity

James Oxley

in Matroid Theory

Published in print:
2011
Published Online:
December 2013
ISBN:
9780198566946
eISBN:
9780191774904
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780198566946.003.0009
Subject:
Mathematics, Educational Mathematics

This chapter addresses the question of whether, for arbitrary n, the property of n-connectedness for graphs can be extended to matroids. It focuses on the case when n = 3. The chapter is organized as ... More


Binary Matroids

James Oxley

in Matroid Theory

Published in print:
2011
Published Online:
December 2013
ISBN:
9780198566946
eISBN:
9780191774904
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780198566946.003.0010
Subject:
Mathematics, Educational Mathematics

This chapter discusses the properties of binary matroids. In particular, it proves numerous characterizations of such matroids. It looks at two vector spaces which are commonly associated with a ... More


Excluded-Minor Theorems

James Oxley

in Matroid Theory

Published in print:
2011
Published Online:
December 2013
ISBN:
9780198566946
eISBN:
9780191774904
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780198566946.003.0011
Subject:
Mathematics, Educational Mathematics

This chapter is organized as follows. Section 10.1 presents Gerards' (1989) proof of Tutte's (1958) excluded-minor characterization of the class of regular matroids. Section 10.2 proves the ... More


The Splitter Theorem

James Oxley

in Matroid Theory

Published in print:
2011
Published Online:
December 2013
ISBN:
9780198566946
eISBN:
9780191774904
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780198566946.003.0013
Subject:
Mathematics, Educational Mathematics

This chapter states and proves Seymour's Splitter Theorem. This result, which is a generalization of Tutte's Wheels-and-Whirls Theorem (8.8.4), is a very powerful general tool for deriving matroid ... More


Factoring 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/9780691175638.003.0007
Subject:
Mathematics, Applied Mathematics

This chapter focuses on Hall's Theorem, introduced by British mathematician Philip Hall, and its connection to graph theory. It first considers problems that ask whether some collection of objects ... More


Graphs, ribbon graphs, and polynomials

Adrian Tanasa

in Combinatorial Physics: Combinatorics, Quantum Field Theory, and Quantum Gravity Models

Published in print:
2021
Published Online:
May 2021
ISBN:
9780192895493
eISBN:
9780191914973
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/oso/9780192895493.003.0002
Subject:
Physics, Theoretical, Computational, and Statistical Physics

In this chapter we present some notions of graph theory that will be useful in the rest of the book. It is worth emphasizing that graph theorists and theoretical physicists adopt, unfortunately, ... More


Fermionic QFT, Grassmann calculus, and combinatorics

Adrian Tanasa

in Combinatorial Physics: Combinatorics, Quantum Field Theory, and Quantum Gravity Models

Published in print:
2021
Published Online:
May 2021
ISBN:
9780192895493
eISBN:
9780191914973
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/oso/9780192895493.003.0006
Subject:
Physics, Theoretical, Computational, and Statistical Physics

In the first section of this chapter, we use Grassmann calculus, used in fermionic QFT, to give, first a reformulation of the Lingström–Gesse–Viennot lemma proof. We further show that this proof ... More


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