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))≤ {reg (I(G)) +2s, reg (I(G)^(s+1)+I(G)^s)}. 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{2, 3, 4}, provided that the complementary graph G is chordal.