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 $G$.