Damped random walks and the characteristic polynomial of the weighted Laplacian on a graph

For λ > 0, we define a λ-damped random walk to be a random walk that is started from a random vertex of a graph and stopped at each step with probability λ 1+λ , otherwise continued with probability 1 1+λ . We use the Aldous-Broder algorithm ([1, 2]) of generating a random spanning tree and the Matrix-tree theorem to relate the values of the characteristic polynomial of the Laplacian at ±λ and the stationary measures of the sets of nodes visited by i independent λ-damped random walks for i ∈ N. As a corollary, we obtain a new characterization of the non-zero eigenvalues of the Weighted Graph Laplacian.