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 $s\geq 1$, $$ \mathrm{reg} (I(G)^{(s+1)})\leq \max \bigl \{\mathrm{reg} (I(G))$$ $$+2s, \mathrm{reg} \bigl (I(G)^{(s+1)}+I(G)^s\bigr )\bigr \}. $$ As a consequence, we conclude that $\mathrm{reg} (I(G)^{(2)})\leq \mathrm{reg} (I(G))+2$, and $\mathrm{reg} (I(G)^{(3)})\leq \mathrm{reg} (I(G))+4$. 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 $\mathrm{reg} (I(G)^{(s)})\leq 2s+\mathrm{reg} (I(G))-2$, for every integer $s\leq k+1$. Finally, it is proven that $\mathrm{reg} (I(G)^{(s)})=2s$, for $s\in \{2, 3, 4\}$, provided that the complementary graph $\overline {G}$ is chordal.