  Coined Quantum Walks (QWs) are being used in many contexts with the goal of
understanding quantum systems and building quantum algorithms for quantum
computers. Alternative models such as Szegedy's and continuous-time QWs were
proposed taking advantage of the fact that quantum theory seems to allow
different quantized versions based on the same classical model, in this case,
the classical random walk. In this work, we show the conditions upon which
coined QWs are equivalent to Szegedy's QWs. Those QW models have in common a
large class of instances, in the sense that the evolution operators are equal
when we convert the graph on which the coined QW takes place into a bipartite
graph on which Szegedy's QW takes place, and vice versa. We also show that the
abstract search algorithm using the coined QW model can be cast into Szegedy's
searching framework using bipartite graphs with sinks.
