Let G be a graph with n vertices and let \(S=\mathbb {K}[x_1,\dots ,x_n]\) be the polynomial ring in n variables over a field \(\mathbb {K}\) . 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.
