## 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 λ_{2}(G) denote the second smallest eigenvalue of the Laplacian of G. If λ_{2}(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.

Original language | English |
---|---|

Pages (from-to) | 555-566 |

Number of pages | 12 |

Journal | Geometric and Functional Analysis |

Volume | 15 |

Issue number | 3 |

DOIs | |

State | Published - Jun 2005 |

Externally published | Yes |

## ASJC Scopus subject areas

- Analysis
- Geometry and Topology