Note on edge irregular reflexive labelings of graphs. Issue 2 (1st August 2019) - Bača, Martin; Irfan, Muhammad; Ryan, Joe; Semaničová-Feňovčíková, Andrea; Tanna, Dushyant
Abstract: For a graph G, an edge labeling f e : E ( G ) → 1, 2, …, k e and a vertex labeling f v : V ( G ) → 0, 2, 4, …, 2 k v are called total k -labeling, where k = max k e, 2 k v . The total k -labeling is called an edge irregular reflexive k -labeling of the graph G, if for every two different edges x y and x ′ y ′ of G, one has w t ( x y ) = f v ( x ) + f e ( x y ) + f v ( y ) ≠ w t ( x ′ y ′ ) = f v ( x ′ ) + f e ( x ′ y ′ ) + f v ( y ′ ) . The minimum k for which the graph G has an edge irregular reflexive k -labeling is called the reflexive edge strength of G . In this paper we determine the exact value of the reflexive edge strength for cycles, Cartesian product of two cycles and for join graphs of the path and cycle with 2 K 2 .