Eigenvalues and homology of flag complexes and vector representations of graphs

Abstract

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 λ_2(G) denote the second smallest eigenvalue of the Laplacian of G. If λ_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 Γ(G) defined via certain vector representations of G. This in turns implies a Hall type theorem for systems of disjoint representatives in hypergraphs.