  We quantitatively differentiate between the spreads of discrete-time quantum
and classical random walks on a cyclic graph. Due to the closed nature of any
cyclic graph, there is additional "collision"- like interference in the quantum
random walk along with the usual interference in any such walk on any graph,
closed or otherwise. We find that the quantum walker remains localized in
comparison to the classical one, even in the absence of disorder, a phenomenon
that is potentially attributable to the additional interference in the quantum
case. This is to be contrasted with the situation on open graphs, where the
quantum walker, being effectively denied the collision-like interference,
garners a much higher spread than its classical counterpart. We use Shannon
entropy of the position probability distribution to quantify spread of the
walker in both quantum and classical cases. We find that for a given number of
vertices on a cyclic graph, the entropy with respect to number of steps for the
quantum walker saturates, on average, to a value lower than that for the
corresponding classical one. We also analyze variations of the entropies with
respect to system size, and look at the corresponding asymptotic growth rates.
