Non-inclusive and inclusive distance irregularity strength for the join product of graphs

Abstract

A function φ : V (G) → {1, 2, . . . , k} of a simple graph G is said to be a non-inclusive distance vertex irregular k-labeling of G if the sums of labels of vertices in the open neighborhood of every vertex are distinct and is said to be an inclusive distance vertex irregular k-labeling of G if the sums of labels of vertices in the closed neighborhood of each vertex are different. The minimum k for which G has a non-inclusive (resp. an inclusive) distance vertex irregular k-labeling is called a non-inclusive (resp. an inclusive) distance irregularity strength and is denoted by dis(G) (resp. by dis( c G)). In this paper, the non-inclusive and inclusive distance irregularity strength for the join product graphs are investigated