Edge Irregularity Strength of Binomial Trees

Abstract

For a simple graph G, a vertex labeling φ: V (G) → {1, 2, · · ·, k} is called k- labeling. The weight of an edge uv in G, denoted by w_φ(uv), is the sum of the labels of end vertices u and v. A vertex k-labeling is defined to be an edge irregular k- labeling of the graph G if for every two different edges e and f, w_φ(e) w_φ(f). The minimum k for which the graph G has an edge irregular k-labeling is called the edge irregularity strength of G, denoted by es(G). In this paper, we prove that the edge irregularity strength of corona product of a tree T with K₁ is es(T K₁) = 2es(T). Further, we prove that the edge irregularity strength of binomial trees B_k is 2^k−1, for k ≥ 1.