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Regularity of powers of edge ideals of unicyclic graphs. (English) Zbl 1417.05093

Summary: Let \(G\) be a finite simple graph and \(I(G)\) denote the corresponding edge ideal. In this paper, we prove that, if \(G\) is a unicyclic graph, then, for all \(s \geq 1\), the regularity of \(I(G)^s\) is exactly \(2s+\mathrm{reg} (I(G))-2\). We also give a combinatorial characterization of unicyclic graphs with regularity \(\nu (G)+1\) and \(\nu (G)+2\), where \(\nu (G)\) denotes the induced matching number of \(G\).

MSC:

05C25 Graphs and abstract algebra (groups, rings, fields, etc.)
05C38 Paths and cycles
05E40 Combinatorial aspects of commutative algebra
13D02 Syzygies, resolutions, complexes and commutative rings
13F20 Polynomial rings and ideals; rings of integer-valued polynomials

Software:

Macaulay2
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References:

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