2021/02/28 by Mohsen Jannesari, Jannesari, Mohsen
Mathematics · #Combinatorics (math.CO) #FOS: Mathematics #math.CO
paper · pdf · doi:10.48550/arxiv.2103.00607
arxiv created 2021/02/28 · arxiv updated 2021/03/02
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.