Perfect state transfer in Grover walks between states associated to vertices of a graph

We study perfect state transfer in Grover walks, which are typical discrete-time quantum walk models. In particular, we focus on states associated to vertices of a graph. call such vertex type states. Perfect between can be studied via Chebyshev polynomials. derive necessary condition eigenvalues graph for occur. addition, perfectly determine the complete multipartite graphs whose partite sets same size occurs states, together with time.