  Let $\mathbb{K}$ be a field and $S=\mathbb{K}[x_1,\dots,x_n]$ be the
polynomial ring in $n$ variables over $\mathbb{K}$. Let $G$ be a graph with $n$
vertices. Assume that $I=I(G)$ is the edge ideal of $G$ and $J=J(G)$ is its
cover ideal. We prove that ${\rm sdepth}(J)\geq n-\nu_{o}(G)$ and ${\rm
sdepth}(S/J)\geq n-\nu_{o}(G)-1$, where $\nu_{o}(G)$ is the ordered matching
number of $G$. We also prove the inequalities ${\rm sdepth}(J^k)\geq {\rm
depth}(J^k)$ and ${\rm sdepth}(S/J^k)\geq {\rm depth}(S/J^k)$, for every
integer $k\gg 0$, when $G$ is a bipartite graph. Moreover, we provide an
elementary proof for the known inequality ${\rm reg}(S/I)\leq \nu_{o}(G)$.
