Eigenvalues and homology of flag complexes and vector representations of graphs

Abstract.

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 $\lambda_{2}(G)\,>\,\frac{k}{k+1}|V|$then $\ifmmode\expandafter\tilde\else\expandafter\~\fi{H}^{k} {\left( {X(G);\mathbb{R}} \right)} = 0.$

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.