Eigenvalues and homology of flag complexes and vector representations of graphs

R. Aharoni, E. Berger, R. Meshulam

Research output: Contribution to journalArticlepeer-review


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 H̃k(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 languageEnglish
Pages (from-to)555-566
Number of pages12
JournalGeometric and Functional Analysis
Issue number3
StatePublished - Jun 2005
Externally publishedYes

ASJC Scopus subject areas

  • Analysis
  • Geometry and Topology


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