  In [S. Arumugam, V. Mathew and J. Shen, On fractional metric dimension of
graphs, preprint], Arumugam et al. studied the fractional metric dimension of
the cartesian product of two graphs, and proposed four open problems. In this
paper, we determine the fractional metric dimension of vertex-transitive
graphs, in particular, the fractional metric dimension of a vertex-transitive
distance-regular graph is expressed in terms of its intersection numbers. As an
application, we calculate the fractional metric dimension of Hamming graphs and
Johnson graphs, respectively. Moreover, we give an inequality for metric
dimension and fractional metric dimension of an arbitrary graph, and determine
all graphs when the equality holds. Finally, we establish bounds on the
fractional metric dimension of the cartesian product of graphs. As a result, we
completely solve the four open problems.
