  We have studied an evolutionary prisoner's dilemma game with players located
on two types of random regular graphs with a degree of 4. The analysis is
focused on the effects of payoffs and noise (temperature) on the maintenance of
cooperation. When varying the noise level and/or the highest payoff, the system
exhibits a second order phase transition from a mixed state of cooperators and
defectors to an absorbing state where only defectors remain alive. For the
random regular graph (and Bethe lattice) the behavior of the system is similar
to those found previously on the square lattice with nearest neighbor
interactions, although the measure of cooperation is enhanced by the absence of
loops in the connectivity structure. For low noises the optimal connectivity
structure is built up from randomly connected triangles.
