Author(s)

Mark Kempton

Center of Mathematical Sciences and Applications, Harvard University, Cambridge, MA, USA.

We study the mixing rate of non-backtracking random walks on graphs by looking at non-backtracking walks as walks on the directed edges of a graph. A result known as Ihara’s Theorem relates the adjacency matrix of a graph to a matrix related to non-backtracking walks on the directed edges. We prove a weighted version of Ihara’s Theorem which relates the transition probability matrix of a non-backtracking walk to the transition matrix for the usual random walk. This allows us to determine the spectrum of the transition probability matrix of a non-backtracking random walk in the case of regular graphs and biregular graphs. As a corollary, we obtain a result of Alon et al. in [1] that in most cases, a non-backtracking random walk on a regular graph has a faster mixing rate than the usual random walk. In addition, we obtain an analogous result for biregular graphs.

Graph, Random Walk, Non-Backtracking Random Walk, Ihara Zeta Identity, Mixing Rate

Kempton, M. (2016) Non-Backtracking Random Walks and a Weighted Ihara’s Theorem. Open Journal of Discrete Mathematics, 6, 207-226. doi: 10.4236/ojdm.2016.64018.