  The spectral theory of random walks on networks of arbitrary topology can be
readily extended to study random walks and L\'evy flights subject to resetting
on these structures. When a discrete-time process is stochastically brought
back from time to time to its starting node, the mean search time needed to
reach another node of the network may be significantly decreased. In other
cases, however, resetting is detrimental to search. Using the eigenvalues and
eigenvectors of the transition matrix defining the process without resetting,
we derive a general criterion for finite networks that establishes when there
exists a non-zero resetting probability that minimizes the mean first passage
time at a target node. Right at optimality, the coefficient of variation of the
first passage time is not unity, unlike in continuous time processes with
instantaneous resetting, but above 1 and depends on the minimal mean first
passage time. The approach is general and applicable to the study of different
discrete-time ergodic Markov processes such as L\'evy flights, where the
long-range dynamics is introduced in terms of the fractional Laplacian of the
graph. We apply these results to the study of optimal transport on rings and
Cayley trees.
