On the regularity of squarefree part of symbolic powers of edge ideals

Assume that G is a graph with edge ideal $I\left(G\right)$. For every integer $s\ge 1$, we denote the squarefree part of the s-th symbolic power of $I\left(G\right)$ by $I{\left(G\right)}^{\left\{s\right\}}$. We determine an upper bound for the regularity of $I{\left(G\right)}^{\left\{s\right\}}$ when G is a chordal graph. If G is a Cameron-Walker graph, we compute $\mathrm{reg}\left(I{\left(G\right)}^{\left\{s\right\}}\right)$ in terms of the induced matching number of G. Moreover, for any graph G, we provide sharp upper bounds for $\mathrm{reg}\left(I{\left(G\right)}^{\left\{2\right\}}\right)$ and $\mathrm{reg}\left(I{\left(G\right)}^{\left\{3\right\}}\right)$.