On the Total Edge Irregularity Strength of Generalized Butterfly Graph

Abstract

Let G(V, E) be a connected, simple, and undirected graph with vertex set V and edge set E. A total k-labeling is a map that carries vertices and edges of a graph G into a set of positive integer labels {1, 2, ⋯, k}. An edge irregular total k-labeling λ: V(G) ∪ E(G) → {1, 2, ⋯, k} of a graph G is a total k-labeling such that the weights calculated for all edges are distinct. The weight of an edge uv in G, denoted by wt(uv), is defined as the sum of the label of u, the label of v, and the label of uv. The total edge irregularity strength of G, denoted by tes(G), is the minimum value of the largest label k over all such edge irregular total k-labelings. A generalized butterfly graph, BFn , obtained by inserting vertices to every wing with assumption that sum of inserting vertices to every wing are same then it has 2n + 1 vertices and 4n - 2 edges. In this paper, we investigate the total edge irregularity strength of generalized butterfly graph, BFn , for n > 2. The result is tes(BFn) = [4n/3].