On the metric dimension of line graphs

Let G be a (di)graph. A set W of vertices in G is a resolving set of G if every vertex u of G is uniquely determined by its vector of distances to all the vertices in W. The metric [email protected](G) of G is the minimum cardinality of all the resolving sets of G. In this paper we study the metric dimension of the line graph L(G) of G. In particular, we show that @m(L(G))=|E(G)|-|V(G)| for a strongly connected digraph G which is not a directed cycle, where V(G) is the vertex set and E(G) is the edge set of G. As a corollary, the metric dimension of de Bruijn digraphs and Kautz digraphs is given. Moreover, we prove that @?log"[email protected](G)@[email protected][email protected](L(G))@?|V(G)|-2 for a simple connected graph G with at least five vertices, where @D(G) is the maximum degree of G. Finally, we obtain the metric dimension of the line graph of a tree in terms of its parameters.