For an ordered set W = (~1, ~2,. , wk} of vertices and a vertex 2) in a connected graph G, the (metric) representation of v with respect to W is the /c-vector T(V 1 W) = (d(v,wl), d(v, wz), , d(v, wk)), where d(z, y) represents the distance between the vertices z and y. The set IV is a resolving set for G if distinct vertices of G have distinct representations.
A new sharp lower bound for the dimension of a graph G in terms of its maximum degree is presented.
A resolving set of minimum cardinality is a basis for G and the number of vertices in a basis is its (metric) dimension dim(G). A resolving set S of G is a minimal resolving set if no proper subset of 5' is a resolving set. The maximum cardinality of a minimal resolving set is the upper dimension dim+(G).
The resolving number res(G) of a connected graph G is the minimum k such that every k-set W of vertices of G is also a resolving set of G. Then 1 5 dim(G) 5 dim+(G) 5 res(G) 5 n -1 for every nontrivial connected graph G of order n. It is shown that dim+(G) = res(G) = TZ -1 if and only if G = Kn, while dim+(G) = res(G) = 2 if and only if G is a path of order at least 4 or an odd cycle. The resolving numbers and upper dimensions of some well-known graphs are determined. It is shown that for every pair a, b of integers with 2 5 a 5 b, there exists a connected graph G with dim(G) = dim+(G) = a and res(G) = b. Also, for every positive integer N, there exists a connected graph G with res(G) -dim+(G) 1 N and dim+(G) -dim(G) 2 N.