Graphs with constant adjacency dimension

For a set W of vertices and a vertex v in a graph G, the k-vector r(2)(v vertical bar W) = (a(G)(v, w(1)), ..., a(G) (v, w(k))) is the adjacency representation of v with respect to W, where W = {w(1), ...,w(k)} and a(G)(x, y) is the minimum of 2 and the distance between the vertices x and y. The set W is an adjacency resolving set for C if distinct vertices of G have distinct adjacency representations with respect to W. The minimum cardinality of an adjacency resolving set for G is its adjacency dimension. It is clear that the adjacency dimension of an n-vertex graph G is between 1 and n - 1. The graphs with adjacency dimension 1 and n-1 are known. All graphs with adjacency dimension 2, and all n-vertex graphs with adjacency dimension n - 2 are studied in this paper. In terms of the diameter and order of G, a sharp upper bound is found for adjacency dimension of G. Also, a sharp lower bound for adjacency dimension of G is obtained in terms of order of G. Using these two hounds, all graphs with adjacency dimension 2, and all n-vertex graphs with adjacency dimension n - 2 are characterized.