A Note on Distance Irregular Labeling of Graphs

Abstract

Let us consider a~simple connected undirected graph G = ( V , E ) . For a~graph G we define a~ k -labeling ϕ : V ( G ) → { 1 , 2 , … , k } to be a~distance irregular vertex k -labeling of the graph G if for every two different vertices u and v of G , one has w t ( u ) ≠ w t ( v ) , where the weight of a~vertex u in the labeling ϕ is w t ( u ) = ∑ v ∈ N ( u ) ϕ ( v ) , where N ( u ) is the set of neighbors of u . The minimum k for which the graph G has a~distance irregular vertex k -labeling is known as distance irregularity strength of G , it is denoted as d i s ( G ) . In this paper, we determine the exact value of the distance irregularity strength of corona product of cycle and path with complete graph of order 1 , friendship graph, Jahangir graph and helm graph. For future research, we suggest some open problems for researchers of the same domain of study.