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

Abstract

The regularity {text {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, where {text {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.