The average search probabilities of discrete-time quantum walks

We study the average probability that a discrete-time quantum walk finds marked vertex on graph. first show that, for regular graph, spectrum of transition matrix is determined by weighted adjacency an augmented then consider search distance and find formula in terms its vertex-deleted subgraph. In particular, any family approaches 1/4 as valency goes to infinity. also present more relaxed criterion, intersection array, this limit be approached graphs.