Abstract
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.
Highlights
A random walk on a graph G is a random process on the vertices of G in which, at each step in the walk, we choose uniformly at random among the neighbors of the current vertex
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
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
Summary
A random walk on a graph G is a random process on the vertices of G in which, at each step in the walk, we choose uniformly at random among the neighbors of the current vertex. We accomplish this via walks on the directed edges of a graph. This gives a new proof of the result of Alon et al concerning the mixing rate of a non-backtracking random walk on a regular graph, and generalizes this result to the class of biregular graphs
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