Integer Smith form via the valence: experience with large sparse matrices from homology
ISSAC '00 Proceedings of the 2000 international symposium on Symbolic and algebraic computation
On parallel block algorithms for exact triangularizations
Parallel Computing
Cycle-free chessboard complexes and symmetric homology of algebras
European Journal of Combinatorics
On the Topology of Discrete Strategies
International Journal of Robotics Research
Fast computation of Smith forms of sparse matrices over local rings
Proceedings of the 37th International Symposium on Symbolic and Algebraic Computation
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Let P be a monotone property of directed graphs on n vertices, and let $\Delta_n^{P}$ denote the abstract simplicial complex whose simplices are the edge sets of graphs having property P. We prove the following: If "P = acyclic,'' then $\Delta_n^{P}$ is homotopy equivalent to the (n-2)-sphere. If "P = not strongly connected,'' then $\Delta_n^{P}$ has the homotopy type of a wedge of (n-1)! spheres of dimension 2n-4. The lattice of all posets on {1,2,...,n} plays an important role in the analysis. We also discuss some other properties of directed graphs from this point of view.