On the Index of Depth Stability of Symbolic Powers of Cover Ideals of Graphs

Let G be a graph with n vertices and let S = K [ x 1 , ⋯ , x n ] be the polynomial ring in n variables over a field K . Assume that I ( G ) and J ( G ) denote the edge ideal and the cover ideal of G , respectively. We provide a combinatorial upper bound for the index of depth stability of symbolic powers of J ( G ). As a consequence, we compute the depth of symbolic powers of cover ideals of fully clique-whiskered graphs. Meanwhile, we determine a class of graphs G with the property that the Castelnuovo–Mumford regularity of S / I ( G ) is equal to the induced matching number of G .