Matching Numbers and the Regularity of the Rees Algebra of an Edge Ideal

Source abstract

Abstract The regularity $${\text {reg}}R(I(G))$$ reg R ( I ( G ) ) of the Rees ring R ( I ( G )) of the edge ideal I ( G ) of a finite simple graph G is studied. It is shown that, if R ( I ( G )) is normal, one has $${\text {mat}}(G) \le {\text {reg}}R(I(G)) \le {\text {mat}}(G) + 1$$ mat ( G ) ≤ reg R ( I ( G ) ) ≤ mat ( G ) + 1 , where $${\text {mat}}(G)$$ mat ( G ) is the matching number of G . In general, the induced matching number is a lower bound for the regularity, which can be shown by applying the squarefree divisor complex.