  For a set W of vertices and a vertex v in a graph G, the k-vector r2(v|W) =
(aG(v,w1),...,aG(v,wk)) is the adjacency representation of v with respect to W,
where W = {w1,...,wk} and aG(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 G 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 bounds, all graphs with adjacency dimension 2, and all n-vertex
graphs with adjacency dimension n-2 are characterized.
