Computing the edge irregularity strengths of chain graphs and the join of two graphs

Abstract

In computer science, graphs are used in variety of applications directly or indirectly. Especially quantitative labeled graphs have played a vital role in computational linguistics, decision making software tools, coding theory and path determination in networks. For a graph G(V, E) with the vertex set V and the edge set E, a vertex k-labeling ϕ : V → {1, 2, …, k} is defined to be an edge irregular k-labeling of the graph G if for every two different edges e and f their w_ϕ(e) ≠ w_ϕ(f), where the weight of an edge e = xy ∈ E(G) is w_ϕ(xy) = ϕ(x) + ϕ(y). The minimum k for which the graph G has an edge irregular k-labeling is called the edge irregularity strength of G, denoted by es(G). In this paper, we determine the edge irregularity strengths of some chain graphs and the join of two graphs. We introduce a conjecture and open problems for researchers for further research.