3 problems

Let $r,d\in\mathbb{Z}_{\ge 0}$, and let $G$ be a connected graph. Write $R$ for the polynomial ring associated with $G$, $J(G)$ for its cover ideal, $\operatorname{reg}(R/J(G))$ fo…

Let $G$ be a connected graph on $n$ vertices. Write $R$ for the polynomial ring associated with $G$, $J(G)$ for its cover ideal, $\operatorname{reg}(R/J(G))$ for the Castelnuovo–Mu…

Let $G$ be a graph with $n$ vertices, let $J(G)$ be its cover ideal, and let $J(G)^{(k)}$ denote its $k$-th symbolic power. Let $\nu_o(G)$ denote the parameter used in the source f…