On total H-irregularity strength of diamond ladder, three circular ladder, and prism graphs

Let G be a graph with vertex set V and edge set E . A total labeling φ : V ( G )∪ E ( G ) → {1, 2, 3, …, α } is called a total α -labeling of a graph G . For the subgraph H ⊆ G under the total α -labeling, H -weight is defined as wt φ ( H ) = ∑ v ∈ V ( H ) φ ( v ) + ∑ e ∈ E ( H ) φ ( e ). A total α -labeling is called an H -irregular total α -labeling of the graph G if wt φ ( H ′) ≠ wt φ ( H ”) for any two distinct subgraphs H ′ and H ” isomorphic to H . 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 total H -irregularity strength of G and we have obtained the tHs of diamond ladder, three circular ladder and prism graphs.