On arithmetic infinite graphs

We compute explicitly the Selberg trace formula for principal congruence subgroups \Gamma of PGL(2, \mathbf{F}_q[t]), which is the modular group in positive characteristic cases. It is known that \Gamma \backslash X is an infinite Ramanujan diagram, where X is the q + 1-regular tres. We express the Selberg zeta function for \Gamma as the determinant of the adjacency operator which is composed of both discrete and continuous spectra. They are rational functions in q^{-s}. We also discuss the limit distribution of eigenvalues of \Gamma \backslash X as the level tends to infinity.