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Let $G$ be a graph with $n$ vertices and let $S=\mathbb{K}[x_1,\dots,x_n]$ bethe 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 depthstability of symbolic powers of $J(G)$. As a consequence, we compute the depthof symbolic powers of cover ideals of fully clique-whiskered graphs. Meanwhile,we determine a class of graphs $G$ with the property that theCastelnuovo--Mumford regularity of $S/I(G)$ is equal to the induced matchingnumber of $G$.

On the index of depth stability of symbolic powers of cover ideals of graphs