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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(More)
Let ∆ n−1 denote the (n − 1)-dimensional simplex. Let Y be a random k-dimensional subcomplex of ∆ n−1 obtained by starting with the full (k − 1)-dimensional skeleton of ∆ n−1 and then adding each k-simplex independently with probability p. Let H k−1 (Y ; R) denote the (k − 1)-dimensional reduced homology group of Y with coefficients in a finite abelian(More)
Let q be a prime power. It is shown that for any hypergraph ~,~ = {F~,..., Fdtq_~)+~ } whose maximal degree is d, there exists Z ¢ ~o c ~, such that IUF~oFI =-0 (rood q). For integers d, m __ 1 let fe(m) denote the minimal t such that for any hypergraph-~ = {Fz ..... Ft} whose maximal degree is d, there exists ~ ¢ o~ o c Y, such that I~F~ ~oFI-= 0 (mod m).(More)
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