The University of Michigan Combinatorics Seminar


Abstract 

To every finite collection of graphs that is closed under removal of edges, one
can associate an abstract simplicial complex whose faces are the edge sets
of the graphs in the collection. Graph complexes have provided an
important link
between combinatorics and algebra, topology and geometry. In
this talk we will discuss the simplicial complex associated with the collection
of subgraphs of a graph G whose maximum vertex degree is at most b. Some
important special cases which have arisen in various contexts in the recent
literature are the matching complex (G is a complete graph and b=1)
and the chessboard complex (G is a complete bipartite graph and b=1).
Topological properties of the matching complex were first examined by Bouc in
connection with Quillen complexes, and topological properties of the
chessboard complex were first examined by Garst in connection with Coxeter
complexes. In this talk I will survey old and new results on the homology
of bounded degree graph complexes. I will discuss the roles played by some
classical symmetric function identities of Littlewood, the representation
theory of symmetric and general linear groups, and the RobinsonSchensted
correspondence.
