On the fractional metric dimension of graphs

In Arumugam et al. (2013), 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 for which 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.