A spectral approach to the shortest path problem

Let G = ( V , E ) be a simple, connected graph. One is often interested in short path between two vertices u v ∈ . We propose spectral algorithm: construct the function ϕ : → R ≥ 0 arg ⁡ min f ≢ ∑ w 1 2 − can also understood as smallest eigenvector of Laplacian Matrix L D A after -th row and column have been removed. start point from to at each step, we move neighbor for which smallest. This algorithm provably terminates results shortest. The efficiency this method due discrete analogue phenomenon Partial Differential Equations that not well understood. prove optimality trees discuss number open questions.