Eigenvalues and homology of flag complexes and vector representations of graphs

Details Eigenvalues and homology of flag complexes and vector representations of graphs Eigenvalues and homology of flag complexes and vector representations of graphs

2003-12-29

arxiv.org/abs/math/0312482v1

Mathematics

Combinatorics

14 pages

Scientific paper

Let X(G) denote the flag complex of a graph G=(V,E) on n vertices. We study relations between the first eigenvalues of successive higher Laplacians of X(G). One consequence is the following result: Let \lambda_2(G) denote the second smallest eigenvalue of the Laplacian of G. If \lambda_2(G)> \frac{kn}{k+1} then the real k-th reduced cohomology group H^k(X(G)) is zero. Applications include a lower bound on the homological connectivity of the independent sets complex I(G), in terms of a new graph domination parameter \Gamma(G) defined via certain vector representations of G. This in turns implies a Hall type theorem for systems of disjoint representatives in hypergraphs.

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