It is known that finitely generated FI-modules over a field of characteristic 0 are Noetherian. We generalize this result to the abstract setting of an infinite EI category satisfying certain combinatorial conditions.

A simplicial complex Δ is called$flag$if all minimal nonfaces of Δ have at most two elements. The following are proved: First, if Δ is a flag simplicial pseudomanifold of dimension$d$−1, then the graph of Δ (i) is (2$d$−2)-vertex-connected and (ii) has a subgraph which is a subdivision of the graph of the$d$-dimensional cross-polytope. Second, the$h$-vector of a flag simplicial homology sphere Δ of dimension$d$−1 is minimized when Δ is the boundary complex of the$d$-dimensional cross-polytope.

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We derive bounds for eigenvalues of the Laplacian of graphs using discrete versions of the Sobolev inequalities and heat kernel estimates.

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