On the Castelnuovo–Mumford regularity of symbolic powers of cover ideals

Assume that [Formula: see text] is a graph with cover ideal [Formula: see text]. For every integer [Formula: see text], we denote the [Formula: see text]th symbolic power of [Formula: see text] by [Formula: see text]. We provide a sharp upper bound for the regularity of [Formula: see text] in terms of the star packing number of [Formula: see text]. Also, for any integer [Formula: see text], we study the difference between [Formula: see text] and [Formula: see text]. As a consequence, we compute the regularity of [Formula: see text] when [Formula: see text] is a doubly Cohen–Macaulay graph. Furthermore, we determine [Formula: see text] if [Formula: see text] is either a Cameron–Walker graph or a claw-free graph which has no cycle of length other than [Formula: see text] and [Formula: see text].