Symbolic powers of edge ideals of graphs

Abstract

Let G be a graph and let I = I(G) be its edge ideal. When G is unicyclic, we give a decomposition of symbolic powers of I in terms of its ordinary powers. This allows us to explicitly compute the Waldschmidt constant and the resurgence number of I. When G is an odd cycle, we explicitly compute the regularity of I(s) for all s ∈ N. In doing so, we also give a natural lower bound for the regularity function reg I(s), for s ∈ N, for an arbitrary graph G.