  We present analytical results for the distribution of first-passage (FP)
times of random walks (RWs) on random regular graphs that consist of $N$ nodes
of degree $c \ge 3$. Starting from a random initial node at time $t=0$, at each
time step $t \ge 1$ an RW hops into a random neighbor of its previous node. In
some of the time steps the RW may hop into a yet-unvisited node while in other
time steps it may revisit a node that has already been visited before. We
calculate the distribution $P( T_{\rm FP} = t )$ of first-passage times from a
random initial node $i$ to a random target node $j$, where $j \ne i$. We
distinguish between FP trajectories whose backbone follows the shortest path
(SPATH) from the initial node $i$ to the target node $j$ and FP trajectories
whose backbone does not follow the shortest path ($\lnot {\rm SPATH}$). More
precisely, the SPATH trajectories from the initial node $i$ to the target node
$j$ are defined as trajectories in which the subnetwork that consists of the
nodes and edges along the trajectory is a tree network. Moreover, the shortest
path between $i$ and $j$ on this subnetwork is the same as in the whole
network. The SPATH scenario is probable mainly when the length $\ell_{ij}$ of
the shortest path between the initial node $i$ and the target node $j$ is
small. The analytical results are found to be in very good agreement with the
results obtained from computer simulations.
