# Regularity of powers of forests and cycles

Research paper by Selvi Beyarslan, Huy Tài Hà, Trân Nam Trung

Indexed on: 14 Jul '15Published on: 14 Jul '15Published in: Journal of Algebraic Combinatorics

#### Abstract

Let G be a graph and let $$I = I(G)$$ be its edge ideal. In this paper, when G is a forest or a cycle, we explicitly compute the regularity of $$I^s$$ for all $$s \ge 1$$. In particular, for these classes of graphs, we provide the asymptotic linear function $${{\mathrm{reg}}}(I^s)$$ as $$s \gg 0$$, and the initial value of s starting from which $${{\mathrm{reg}}}(I^s)$$ attains its linear form. We also give new bounds on the regularity of I when G contains a Hamiltonian path and when G is a Hamiltonian graph.