  Let $G$ be a graph with $n$ vertices and $S=\mathbb{K}[x_1,\dots,x_n]$ be the
polynomial ring in $n$ variables over a field $\mathbb{K}$. Assume that $J(G)$
is the cover ideal of $G$ and $J(G)^{(k)}$ is its $k$-th symbolic power. We
prove that if $G$ is a very well-covered graph such that $J(G)$ has linear
resolution, then $J(G)^{(k)}$ has linear resolution, for every integer $k\geq
1$. We also prove that for a every very well-covered graph $G$, the depth of
symbolic powers of $J(G)$ forms a non-increasing sequence. Finally, we
determine a linear upper bound for the regularity of powers of cover ideal of
bipartite graph.
