  We present analytical results for the distribution of first return (FR) times
of random walks (RWs) on random regular graphs (RRGs) consisting of $N$ nodes
of degree $c \ge 3$. Starting from a random initial node $i$ at time $t=0$, at
each time step $t \ge 1$ an RW hops into a random neighbor of its previous
node. We calculate the distribution $P ( T_{\rm FR} = t )$ of first return
times to the initial node $i$. We distinguish between first return trajectories
in which the RW retrocedes its own steps backwards all the way back to the
initial node $i$ and those in which the RW returns to $i$ via a path that does
not retrocede its own steps. In the retroceding scenario, each edge that
belongs to the RW trajectory is crossed the same number of times in the forward
and backward directions. In the non-retroceding scenario the subgraph that
consists of the nodes visited by the RW and the edges it has crossed between
these nodes includes at least one cycle. In the limit of $N \rightarrow \infty$
the RRG converges towards the Bethe lattice. The Bethe lattice exhibits a tree
structure, in which all the first return trajectories belong to the retroceding
scenario. Moreover, in the limit of $N \rightarrow \infty$ the trajectories of
RWs on RRGs are transient in the sense that they return to the initial node
with probability $<1$. In this sense they resemble the trajectories of RWs on
regular lattices of dimensions $d \ge 3$. The analytical results are found to
be in excellent agreement with the results obtained from computer simulations.
