On the regularity of small symbolic powers of edge ideals of graphs - Seyed Fakhari, S. A. (MATHEMATICA SCANDINAVICA, 2023)
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 $s≥ 1$, $$ reg (I(G)^{(s+1)})≤ \max \bigl \{reg (I(G))$$ $$+2s, reg \bigl (I(G)^{(s+1)}+I(G)^s\bigr )\bigr \}. $$ As a consequence, we conclude that $reg (I(G)^{(2)})≤ reg (I(G))+2$, and $reg (I(G)^{(3)})≤ reg (I(G))+4$. Moreover, it is shown that if for some integer $k≥ 1$, the graph $G$ has no odd cycle of length at most $2k-1$, then $reg (I(G)^{(s)})≤ 2s+reg (I(G))-2$, for every integer $s≤ k+1$. Finally, it is proven that $reg (I(G)^{(s)})=2s$, for $s\in \{2, 3, 4\}$, provided that the complementary graph $\overline {G}$ is chordal.