Note on edge irregular reflexive labelings of graphs

For a graph <i>G</i>, an edge labeling <i>f_e : E(G)</i> → {1, 2, . . . , <i>k_e</i>} and a vertex labeling <i>f_v : V(G</i>) → {0, 2, 4, . . . , 2<i>k_v</i>} are called total <i>k</i>-labeling, where <i>k</i> = max{<i>k_e</i>, 2<i>k_v</i>}. The total <i>k</i>-labeling is called an <i>edge irregular reflexive k-labeling</i> of the graph <i>G</i>, if for every two different edges <i>xy</i> and <i>x′ y′</i> of G, one has <i>wt(xy)</i> = <i>f_v(x)</i> + <i>f_e(xy)</i> + <i>f_v(y</i>) ̸= <i>wt(x′ y′)</i> = <i>f_v(x′)</i> + <i>f_e(x′ y′)</i> + <i>f_v(y′)</i>. The minimum <i>k</i> for which the graph <i>G</i> has an edge irregular reflexive <i>k</i>-labeling is called the <i>reflexive edge strength of G</i>. 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<i>K</i>₂.