  Continuous-time open quantum walks (CTOQW) are introduced as the formulation
of quantum dynamical semigroups of trace-preserving and completely positive
linear maps (or quantum Markov semigroups) on graphs. We show that a CTOQW
always converges to a steady state regardless of the initial state when a graph
is connected. When the graph is both connected and regular, it is shown that
the steady state is the maximally mixed state. As shown by the examples in this
article, the steady states of CTOQW can be very unusual and complicated even
though the underlying graphs are simple. The examples demonstrate that the
structure of a graph can affect quantum coherence in CTOQW through a long time
run. Precisely, the quantum coherence persists throughout the evolution of the
CTOQW when the underlying topology is certain irregular graphs (such as a path
or a star as shown in the examples). In contrast, the quantum coherence will
eventually vanish from the open quantum system when the underlying topology is
a regular graph (such as a cycle).
