On the regularity of small symbolic powers of edge ideals of graphs

Assume that $G$ is a graph with edge ideal $I(G)$ and let $I(G)^{(s)}$ denote the $s$-th symbolic power of $I(G)$. It is proved that for every integer $s\geq 1$, $${\rm reg}(I(G)^{(s+1)})\leq \max\bigg\{{\rm reg}(I(G))+2s, {\rm reg}\big(I(G)^{(s+1)}+I(G)^s\big)\bigg\}.$$As a consequence, we conclude that ${\rm reg}(I(G)^{(2)})\leq {\rm reg}(I(G))+2$, and ${\rm reg}(I(G)^{(3)})\leq {\rm reg}(I(G))+4$. Moreover, it is shown that if for some integer $k\geq 1$, the graph $G$ has no odd cycle of length at most $2k-1$, then ${\rm reg}(I(G)^{(s)})\leq 2s+{\rm reg}(I(G))-2$, for every integer $s\leq k+1$. Finally, it is proven that ${\rm reg}(I(G)^{(s)})=2s$, for $s\in \{2, 3, 4\}$, provided that the complementary graph $\overline{G}$ is chordal.