  We systematically study the localization effect in discrete-time quantum
walks on a honeycomb network and establish the mathematical framework. We focus
on the Grover walk first and rigorously derive the limit form of the walker's
state, showing it has a certain probability to be localized at the starting
position. The relationship between localization and the initial coin state is
concisely represented by a linear map. We also define and calculate the average
probability of localization by generating random initial states. Further, coin
operators varying with positions are considered and the sufficient condition
for localization is discussed. We also similarly analyze another four-state
Grover walk. Theoretical predictions are all in accord with numerical
simulation results. Finally, our results are compared with previous works to
demonstrate the unusual trapping effect of quantum walks on a honeycomb
network, as well as the advantages of our method.
