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Matroids : a geometric introduction

Author: Gary Gordon; Jennifer McNulty
Publisher: Cambridge ; New York : Cambridge University Press, 2012.
Edition/Format:   Book : EnglishView all editions and formats
Database:WorldCat
Summary:

This friendly introduction helps undergraduate students understand and appreciate matroid theory and its connections to geometry.

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Details

Document Type: Book
All Authors / Contributors: Gary Gordon; Jennifer McNulty
ISBN: 9780521767248 0521767245 9780521145688 0521145686
OCLC Number: 779244506
Description: xv, 393 p. : ill. ; 26 cm.
Contents: 1. A tour of matroids --
1.1. Motivation --
1.2. Introduction to matroids --
1.3. Geometries --
1.4. Graphs and matroids --
1.5. Bipartite graphs and transversal matroids --
Exercises --
2. Cryptomorphisms --
2.1. From independent sets to bases and back again --
2.2. Circuits and independent sets --
2.3. Rank, flats, hyperplanes and closure --
2.4. Lattice of flats --
2.5. Tying it together with the rank function --
2.6. Cryptomorphisms between flats, hyperplanes and closure --
2.7. Application to optimization: the greedy algorithm --
Exercises --
3. New matroids from old --
3.1. Matroid deletion and contraction --
3.2. Deletion and contraction in graphs and representable matroids --
3.3. Duality in matroids --
3.4. Duality in graphic and representable matroids --
3.5. Direct sums and connectivity --
Exercises --
4. Graphic matroids --
4.1. Graphs are matroids --
4.2. Graph versions of matroid theorems --
4.3. Duality and cocircuits in graphs --
4.4. Connectivity and 2-isomorphism --
Exercises --
5. Finite geometry --
5.1. Affine geometry and affine dependence --
5.2. Affine dependence --
5.3. The projective plane PG(2, q) --
5.4. Projective geometry --
5.5. Counting k-flats and q-binomial coefficients --
5.6. Abstract projective planes --
Exercises --
6. Representable matroids --
6.1. Matrices are matroids --
6.2. Representing representable matroids --
6.3. How representations depend on the field --
6.4. Non-representable matroids --
6.5. Representations and geometry --
Exercises --
7. Other matroids --
7.1. Transversal matroids --
7.2. Transversal matroids, matching theory and geometry --
7.3. Hyperplane arrangements and matroids --
7.4. Cutting cheese; counting regions via deletion and contraction --
Exercises --
8. Matroid minors --
8.1. Examples, excluded minors and the Scum Theorem --8.2. Binary matroids --
8.3. Summary of excluded minor results --
Exercises --
9. The Tutte polynomial --
9.1. Motivation and history --
9.2. Definition and basic examples --
9.3. Corank--nullity polynomial --
9.4. Duality and direct sum --
9.5. Tutte--Grothendieck invariants --
9.6. The chromatic polynomial --
Exercises --
Projects --
P.1. The number of matroids --
P.2. Matrix-Tree Theorem --
P.3. Relaxing a hyperplane --
P.4. Bases and probability in affine and projective space --
P.5. Representing affine space --
the characteristic set of AG(n, q) --
P.6. The card game SET® and affine geometry --
P.7. More matroid constructions --
truncation --
Appendix Matroid axiom systems --
A.1. Axiom lists --
A.2. Axiom tables.
Responsibility: Gary Gordon, Jennifer McNulty.
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"The authors write in an entertaining, conversational style, and the text is often peppered with humorous footnotes. Nearly 300 exercises and scores of references will benefit motivated readers." -J. Read more...

 
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