# Symbolic powers of edge ideals of graphs

Research paper by Yan Gu, Huy Tai Ha, Jonathan L. O'Rourke, Joseph W. Skelton

Indexed on: 09 May '18Published on: 09 May '18Published in: arXiv - Mathematics - Commutative Algebra

#### 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, the resurgence number, and the symbolic defect for $I$. When $G$ is an odd cycle, we explicitly compute the regularity of $I^{(s)}$ for all $s \in \mathbb{N}$. In doing so, we also give a natural lower bound for the regularity function $\text{reg } I^{(s)}$, for $s \in \mathbb{N}$, for an arbitrary graph $G$.