  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 $G$.
