Eigenvalues and homology of flag complexes and vector representations of graphs

The flag complex of a graph G = (V, E) is the simplicial complex X(G) on the vertex set V whose simplices are subsets of V which span complete subgraphs of G. We study relations between the first eigenvalues of successive higher Laplacians of X(G). One consequence is the following: Theorem: Let λ₂(G) denote the second smallest eigenvalue of the Laplacian of G. If λ₂(G) > k/k+1|V| then H̃^k(X(G) ;ℝ) = 0. Applications include a lower bound on the homological connectivity of the independent sets complex I(G), in terms of a new graph domination parameter Γ(G) defined via certain vector representations of G. This in turns implies Hall type theorems for systems of disjoint representatives in hypergraphs.