Graphs with constant adjacency dimension

TL;DR

This paper investigates the adjacency dimension of graphs, providing bounds and characterizations for graphs with adjacency dimensions 2 and n-2, based on diameter and order.

Contribution

It introduces sharp bounds for adjacency dimension and characterizes all graphs with adjacency dimensions 2 and n-2.

Findings

Sharp upper and lower bounds for adjacency dimension based on graph diameter and order.

Complete characterization of graphs with adjacency dimension 2.

Complete characterization of n-vertex graphs with adjacency dimension n-2.

Abstract

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.