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Let $\mathbb{K}$ be a field and $S=\mathbb{K}[x_1,\dots,x_n]$ be thepolynomial 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 itscover ideal. We prove that ${\rm sdepth}(J)\geq n-\nu_{o}(G)$ and ${\rmsdepth}(S/J)\geq n-\nu_{o}(G)-1$, where $\nu_{o}(G)$ is the ordered matchingnumber of $G$. We also prove the inequalities ${\rm sdepth}(J^k)\geq {\rmdepth}(J^k)$ and ${\rm sdepth}(S/J^k)\geq {\rm depth}(S/J^k)$, for everyinteger $k\gg 0$, when $G$ is a bipartite graph. Moreover, we provide anelementary proof for the known inequality ${\rm reg}(S/I)\leq \nu_{o}(G)$.

Depth, Stanley depth and regularity of ideals associated to graphs