  We study quantum Markov chains on graphs, described by completely positive
maps, following the model due to S. Gudder (J. Math. Phys. 49, 072105, 2008)
and which includes the dynamics given by open quantum random walks as defined
by S. Attal et al. (J. Stat. Phys. 147:832-852, 2012). After reviewing such
structures we examine a quantum notion of mean time of first visit to a chosen
vertex. However, instead of making direct use of the definition as it is
usually done, we focus on expressions for such quantity in terms of generalized
inverses associated with the walk and most particularly the so-called
fundamental matrix. Such objects are in close analogy with the theory of Markov
chains and the methods described here allow us to calculate examples that
illustrate similarities and differences between the quantum and classical
settings.
