The Limit Theorem with Respect to the Matrices on Non-backtracking Paths of a Graph

We give a limit theorem with respect to the matrices related to non-backtracking paths of a regular graph. The limit obtained closely resembles the k th moments of the arcsine law. Furthermore, we obtain the asymptotics of the averages of the p m th Fourier coefficients of the cusp forms related to the Ramanujan graphs defined by A. Lubotzky, R. Phillips and P. Sarnak.