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 -labeling is said to be a total edge irregular -labeling of the graph G if for every two di erent edges e 1 and e 2 , it holds w(e 1 ) 6 = w(e ), where w(uv) = f(u) +f(uv) +f(v), for e = uv. The minimum for which the graph G has a total edge irregular -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 -labeling into a total H-irregular -labeling. The total -labeling is said to be a total H-irregular -labeling of the graph G if for H G, the total H-weights W(H) = P 2 v2V (H) f(v) + P f(e) are distinct. The minimum for which the graph G has a total H-irregular -labeling is called the total e2E(H) 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.