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.
Categories: math.AC math.CO