Regularity of Powers of Edge Ideals of Some Graphs

Abstract

Let $\tilde {C_{n}}$ be the graph by adding an ear to C _n and $I=I(\tilde {C_{n}})$ be its edge ideal. In this paper, we prove that $\operatorname {reg}(I^{s})=2s+\lfloor \frac {n+1}{3}\rfloor -1$ for all s ≥ 1. Let G be the bicyclic graph C _m ⊔ C _n with edge ideal I = I(G); we compute the regularity of I ^s for all s ≥ 1. In particular, in some cases, we get $\operatorname {reg}(I^{s})=2s+\lfloor \frac {m}{3}\rfloor +\lfloor \frac {n}{3}\rfloor -1$ for all s ≥ 2.