Exact Grounded Resistance Sums on Path Graphs: Closed Forms, Optimal Landmarks, Cycles, and Edge Sensitivity

We derive an exact closed form for the degree-weighted sum of grounded effective resis-tances on a path graph, and extend it in four directions. First, the closed form yields O(1)evaluation of the effective degrees of freedom for graph-based smoothing and its derivative,enabling fast generalized cross-validation. Second, the Nyström landmark selection problemon a path is solved exactly by dynamic programming, using two new O(1) sub-path costfunctions. Third, the analogous resistance sum for cycle graphs is expressed in closed form.Fourth, the sensitivity of the total resistance to the removal of a single edge is obtained,giving an O(n) vulnerability scan. All results are verified numerically.