On the total H-irregularity strength of graphs: A new notion

Abstract

A total edge irregularity strength of G has been already widely studied in many papers. The total α-labeling is said to be a total edge irregular α-labeling of the graph G if for every two different edges e1 and e2, it holds w(e1) ≠ w(e2), where w(uv) = f(u) + f(uv) + f(v), for e = uv. The minimum α for which the graph G has a total edge irregular α-labeling is called the total edge irregularity strength of G, denoted by tes(G). A natural extension of this concept is by considering the evaluation of the weight is not only for each edge but we consider the weight on each subgraph H ⊆ G. We extend the notion of the total α-labeling into a total H-irregular α-labeling. The total α-labeling is said to be a total H-irregular α-labeling of the graph G if for H ⊆ G, the total H-weights W (H) = ∑v∈V(H) f(v) + ∑e∈E(H) f(e) are distinct. The minimum α for which the graph G has a total H-irregular α-labeling is called the total H-irregularity strength of G, denoted by tHs(G). In this paper we initiate to study the tHs of shackle and amalgamation of any graphs and their bound.