A Note on Edge Irregularity Strength of Some Graphs

Abstract

Let G(V, E) be a finite simple graph and k be some positive integer. A vertex k-labeling of graph G(V,E), Φ : V → {1,2,..., k}, is called edge irregular k-labeling if the edge weights of any two different edges in G are distinct, where the edge weight of e = xy ∈ E(G), w_Φ(e), is defined as w_Φ(e) = Φ(x) + Φ(y). The edge irregularity strength for graph G is the minimum value of k such that Φ is irregular edge k-labeling for G. In this note we derive the edge irregularity strength of chain graphs mK₃−path for m ≢ 3 (mod4) and C[C_n^(m)] for all positive integers n ≡ 0 (mod 4) 3n and m. We also propose bounds for the edge irregularity strength of join graph P_m + Ǩ_n for all integers m, n ≥ 3.