A Note on Edge Irregularity Strength of Some Graphs

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  mK 3 −path for m ≢ 3 (mod4) and C [ C n ( m ) ] for all positive integers  n  ≡ 0 (mod 4) 3 n and  m . We also propose bounds for the edge irregularity strength of join graph  P m  +  Ǩ n  for all integers  m, n  ≥ 3.