4 problems

For positive integers $m,n$, let $P_k$ denote the path on $k$ vertices, and let $P_m\times P_n$ be their Cartesian product, with vertices $(g,h)$ and $(g',h')$ adjacent exactly whe…

Let $G$ be a graph with vertex set $V$, let $c$ be normalized edge weights, and let $\mathcal{S}(G)\subseteq[0,1]^{2^V}$ be the space of resistance capacities. A set function on…

Let $P_n$ and $P_m$ be path graphs, and let $P_n\times P_m$ denote their Cartesian product, a grid graph. A graph is resistance nonnegative (RN) if its resistance curvature is nonn…

Let $G$ be a graph. A graph is 1-tough if deleting any vertex set $S$ leaves at most $|S|$ connected components, and it is resistance positive (RP) if its resistance curvature is p…