Reflexive Edge Strength on Slanting Ladder Graph and Corona of Centipede and Null Graph

Abstract

Assume G is a graph that is simple, undirected, and connected. If every edge label is a positive integer in the range 1 to k_e, and every vertex label is a non-negative even number from 0 to 2k_v, then a graph G is considered to have an edge irregular reflexive k-labeling, where k is defined as the maximum of k_e and 2k_v. The edge weight w_t(ab) in the graph G, for the labeling λ, is defined as the function w_t applied to the edge ab. The symbol res(G) denotes the reflexive edge strength, which is the largest label of the smallest k. The results of this research are as follows: res(SL_m) for m≥2 is ⌈(3m−3)/3⌉ for 3m−3 ≢ 2, 3 (mod 6), and ⌈(3m−3)/3⌉+1 for 3m−3 ≡ 2, 3 (mod 6). res(Cp_n ⊙ N_m) for n≥2, m≥1 is ⌈(2nm+2n−1)/3⌉ for 2nm+2n−1 ≢ 2, 3 (mod 6), and ⌈(2nm+2n−1)/3⌉+1 for 2nm+2n−1 ≡ 2, 3 (mod 6).