  We define a general model of stochastically-evolving graphs, namely the
\emph{Edge-Uniform Stochastically-Evolving Graphs}. In this model, each
possible edge of an underlying general static graph evolves independently being
either alive or dead at each discrete time step of evolution following a
(Markovian) stochastic rule. The stochastic rule is identical for each possible
edge and may depend on the past $k \ge 0$ observations of the edge's state. We
examine two kinds of random walks for a single agent taking place in such a
dynamic graph: (i) The \emph{Random Walk with a Delay} (\emph{RWD}), where at
each step the agent chooses (uniformly at random) an incident possible edge,
i.e., an incident edge in the underlying static graph, and then it waits till
the edge becomes alive to traverse it. (ii) The more natural \emph{Random Walk
on what is Available} (\emph{RWA}) where the agent only looks at alive incident
edges at each time step and traverses one of them uniformly at random. Our
study is on bounding the \emph{cover time}, i.e., the expected time until each
node is visited at least once by the agent. For \emph{RWD}, we provide a first
upper bound for the cases $k = 0, 1$ by correlating \emph{RWD} with a simple
random walk on a static graph. Moreover, we present a modified electrical
network theory capturing the $k = 0$ case. For \emph{RWA}, we derive some first
bounds for the case $k = 0$, by reducing \emph{RWA} to an \emph{RWD}-equivalent
walk with a modified delay. Further, we also provide a framework, which is
shown to compute the exact value of the cover time for a general family of
stochastically-evolving graphs in exponential time. Finally, we conduct
experiments on the cover time of \emph{RWA} in Edge-Uniform graphs and compare
the experimental findings with our theoretical bounds.
