Note on edge irregular reflexive labelings of graphs

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, . . . , 2k_v} are called total k-labeling, where k = max{k_e, 2k_v}. The total k-labeling is called an edge irregular reflexive k-labeling of the graph G, if for every two different edges xy and x′ y′ of G, one has wt(xy) = f_v(x) + f_e(xy) + f_v(y) ̸= wt(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 2K₂.