  Real networks are often dynamic. In response to it, analyses of algorithms on
{\em dynamic networks} attract more and more attentions in network science and
engineering. Random walks on dynamic graphs also have been investigated
actively in more than a decade, where in most cases the edge set changes but
the vertex set is static. The vertex sets are also dynamic in many real
networks. Motivated by a new technology of the analysis of random walks on
dynamic graphs, this paper introduces a simple model of graphs with increasing
the number of vertices, and presents an analysis of random walks associated
with the cover time on such graphs. In particular, we reveal that a random walk
asymptotically covers the vertices all but a constant number if the vertex set
grows {\em moderately}.
