© 2008

Simplicial Complexes of Graphs


Part of the Lecture Notes in Mathematics book series (LNM, volume 1928)

Table of contents

  1. Front Matter
    Pages i-xiv
  2. Introduction and Basic Concepts

  3. Tools

    1. Pages 51-66
    2. Pages 67-86
    3. Pages 87-95
  4. Overview of Graph Complexes

    1. Pages 99-106
    2. Pages 113-118
  5. Vertex Degree

    1. Pages 127-149
    2. Pages 151-168
  6. Cycles and Crossings

  7. Connectivity

    1. Pages 245-262
    2. Pages 263-273

About this book


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.


Graph Homotopy Hypergraph Matching Sim Vertex homology

Authors and affiliations

  1. 1.Department of MathematicsKTHStockholmSweden

Bibliographic information

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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)