One edge. Every mode moves.
Turn route times into conductances, add an endpoint shortcut as a rank-one positive-semidefinite update, and inspect the full weighted Laplacian spectrum.
min
min
min
min
Ready to compute the representative shortcut.
w = 1/t
L' = L + wbb^T
b = [1, 0, 0, -1]^T
L' = L + wbb^T
b = [1, 0, 0, -1]^T
Synthetic four-station graph only. It teaches the stated spectral rule; it is not a live London route map or journey planner.
λ2 before0.154292
λ2 after0.446731
Connectivity gain189.54%
Weyl checkPASS
Ordered eigenvalues
ModeBeforeAfterShift
Original L
Updated L'
The 2.00-minute shortcut adds conductance 0.500000 and raises algebraic connectivity from 0.154292 to 0.446731. Every ordered eigenvalue weakly increases within numerical tolerance.