Simplicial Complexes of Graphs: 1928 2008th Edition

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)

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.