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    Regularity of Edge Ideals and Their Powers

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    We survey recent studies on the Castelnuovo-Mumford regularity of edge ideals of graphs and their powers. Our focus is on bounds and exact values of  reg I(G)\text{ reg } I(G) and the asymptotic linear function  reg I(G)q\text{ reg } I(G)^q, for q≥1,q \geq 1, in terms of combinatorial data of the given graph G.G.Comment: 31 pages, 15 figure

    Perfect Elimination Orderings for Symmetric Matrices

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    We introduce a new class of structured symmetric matrices by extending the notion of perfect elimination ordering from graphs to weighted graphs or matrices. This offers a common framework capturing common vertex elimination orderings of monotone families of chordal graphs, Robinsonian matrices and ultrametrics. We give a structural characterization for matrices that admit perfect elimination orderings in terms of forbidden substructures generalizing chordless cycles in graphs.Comment: 16 pages, 3 figure

    Clique complexes and graph powers

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    We study the behaviour of clique complexes of graphs under the operation of taking graph powers. As an example we compute the clique complexes of powers of cycles, or, in other words, the independence complexes of circular complete graphs.Comment: V3: final versio

    Regularity of squarefree monomial ideals

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    We survey a number of recent studies of the Castelnuovo-Mumford regularity of squarefree monomial ideals. Our focus is on bounds and exact values for the regularity in terms of combinatorial data from associated simplicial complexes and/or hypergraphs.Comment: 23 pages; survey paper; minor changes in V.
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