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Simplicial Complexes of Graphs

(Autor)

Buch | Softcover
XIV, 382 Seiten
2007 | 2008
Springer Berlin (Verlag)
978-3-540-75858-7 (ISBN)

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Simplicial Complexes of Graphs - Jakob Jonsson
CHF 97,35 inkl. MwSt

A graph complex is a finite family of graphs closed under deletion of edges. Graph complexes show up naturally in many different areas of mathematics, including commutative algebra, geometry, and knot theory. Identifying each graph with its edge set, one may view a graph complex as a simplicial complex and hence interpret it as a geometric object. This volume examines topological properties of graph complexes, focusing on homotopy type and homology.

Many of the proofs are based on Robin Forman's discrete version of Morse theory. As a byproduct, this volume also provides a loosely defined toolbox for attacking problems in topological combinatorics via discrete Morse theory. In terms of simplicity and power, arguably the most efficient tool is Forman's divide and conquer approach via decision trees; it is successfully applied to a large number of graph and digraph complexes.

and Basic Concepts.- and Overview.- Abstract Graphs and Set Systems.- Simplicial Topology.- Tools.- Discrete Morse Theory.- Decision Trees.- Miscellaneous Results.- Overview of Graph Complexes.- Graph Properties.- Dihedral Graph Properties.- Digraph Properties.- Main Goals and Proof Techniques.- Vertex Degree.- Matchings.- Graphs of Bounded Degree.- Cycles and Crossings.- Forests and Matroids.- Bipartite Graphs.- Directed Variants of Forests and Bipartite Graphs.- Noncrossing Graphs.- Non-Hamiltonian Graphs.- Connectivity.- Disconnected Graphs.- Not 2-connected Graphs.- Not 3-connected Graphs and Beyond.- Dihedral Variants of k-connected Graphs.- Directed Variants of Connected Graphs.- Not 2-edge-connected Graphs.- Cliques and Stable Sets.- Graphs Avoiding k-matchings.- t-colorable Graphs.- Graphs and Hypergraphs with Bounded Covering Number.- Open Problems.- Open Problems.

From the reviews:

"The subject of this book is the topology of graph complexes. A graph complex is a family of graphs ... which is closed under deletion of edges. ... Topological and enumerative properties of monotone graph properties such as matchings, forests, bipartite graphs, non-Hamiltonian graphs, not-k-connected graphs are discussed. ... Researchers, who find any of the stated problems intriguing, will be enticed to read the book." (Herman J. Servatius, Zentralblatt MATH, Vol. 1152, 2009)

Erscheint lt. Verlag 15.11.2007
Reihe/Serie Lecture Notes in Mathematics
Zusatzinfo XIV, 382 p. 34 illus.
Verlagsort Berlin
Sprache englisch
Maße 155 x 235 mm
Gewicht 602 g
Themenwelt Mathematik / Informatik Mathematik Algebra
Mathematik / Informatik Mathematik Geometrie / Topologie
Mathematik / Informatik Mathematik Wahrscheinlichkeit / Kombinatorik
Schlagworte 05E25, 55U10, 06A11 • Decision Tree • discrete Morse theory • Graph • Homology • Homotopy • Hypergraph • Matching • monotone graph property • SiM • simplicial complex • simplicial homology • Vertex
ISBN-10 3-540-75858-5 / 3540758585
ISBN-13 978-3-540-75858-7 / 9783540758587
Zustand Neuware
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