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.

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We construct infinite families of graphs in which pretty good state transfer can be induced by adding a potential to the nodes of the graph (i.e. adding a number to a diagonal entry of the adjacency matrix). Indeed, we show that given any graph with a pair of cospectral nodes, a simple modification of the graph, along with a suitable potential, yields pretty good state transfer (i.e. asymptotically perfect state transfer) between the nodes. This generalizes previous work, concerning graphs with an involution, to asymmetric graphs.where X i , Y i , Z i are the standard Pauli matrices, J ij denotes the strength of the XX coupling between node i and j, and the Q i 's give the strength of the magnetic field yielding an energy potential at each node.

Two emerging topics in graph theory are the study of cospectral vertices of a graph, and the study of isospectral reductions of graphs. In this paper, we prove a fundamental relationship between these two areas, which is that two vertices of a graph are cospectral if and only if the isospectral reduction over these vertices has a nontrivial automorphism. It is well known that if two vertices of a graph are symmetric, i.e. if there exists a graph automorphism permuting these two vertices, then they are cospectral. This paper extends this result showing that any two cospectral vertices are symmetric in some reduced version of the graph. We also prove that two vertices are strongly cospectral if and only if they are cospectral and the isospectral reduction over these two vertices has simple eigenvalues. We further describe how these results can be used to construct new families of graphs with cospectral vertices. *