Symbolic powers of edge ideals of graphs

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 $s\in \mathbb{N}$. In doing so, we also give a natural lower bound for the regularity function $$\text{reg }$ I^{(s)}$,for$s \in $\mathbb{N}$$,foranarbitrarygraph$G$.