On the Castelnuovo-Mumford regularity of symbolic powers of cover ideals

Assume that $G$ is a graph with cover ideal $J(G)$. For every integer $k\ge 1$, we denote the $k$-th symbolic power of $J(G)$ by $J(G{)}^{(k)}$. We provide a sharp upper bound for the regularity of $J(G{)}^{(k)}$ in terms of the star packing number of $G$. Also, for any integer $k\ge 2$, we study the difference between $\mathrm{r}\mathrm{e}\mathrm{g}(J(G{)}^{(k)})$ and $\mathrm{r}\mathrm{e}\mathrm{g}(J(G{)}^{(k-2)})$. As a consequence, we compute the regularity of $J(G{)}^{(k)}$ when $G$ is a doubly Cohen-Macaulay graph. Furthermore, we determine $\mathrm{r}\mathrm{e}\mathrm{g}(J(G{)}^{(k)})$ if $G$ is either a Cameron-Walker graph or a claw-free graph which has no cycle of length other that $3$ and $5$.