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

Abstract

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 $\fracλ{1+λ}$, otherwise continued with probability $\frac{1}{1+λ}$. We use the Aldous-Broder algorithm (\cite{aldous, broder}) of generating a random spanning tree and the Matrix-tree theorem to relate the values of the characteristic polynomial of the Laplacian at $\pm λ$ and the stationary measures of the sets of nodes visited by $i$ independent $λ$-damped random walks for $i \in \N$. As a corollary, we obtain a new characterization of the non-zero eigenvalues of the Weighted Graph Laplacian.