Branching annihilating random walk on random regular graphs

The branching annihilating random walk is studied on a random graph whose sites have uniform number of neighbors (z). The Monte Carlo simulations in agreement with the generalized mean-field analysis indicate that the concentration decreses linearly with the branching rate for $z \ge 4$ while the coefficient of the linear term becomes zero if $z=3$. These features are described by a modified mieb-field theory taking explicitly into consideration the probability of mutual annihilation of the parent and its offspring particles using the returning features of a single walker on the same graph.