  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.
