Super Edge Bimagic and Trimagic Total Labeling for some Graphs.

A super edge trimagic total labeling (SETTL) of a graph Γ with α vertices and β edges is a bijection Φ:[V (Γ) ∪ E(Γ)] ⟶{1,2,3,· · · ,α+β} such that for each edge ϑω∈E(Γ), the value of the formula [ Φ(ϑ)+Φ(ω)+Φ(ϑω)] is either Κ_1 or Κ_2 or Κ_3, with the additional condition that Φ:[V (Γ)] ⟶{1,2,3,· · · ,α}. A super edge trimagic total graph is the one that allows a super edge trimagic total labeling. The idea of super bimagic total labelling of connected graphs gets further investigated in this study. First, we present the triangulated prism graph 〖TΠ〗_r and demonstrate its bimagic total labelling by using the bimagic numbers K_1=6r and K_2=8r , showing that this graph admits a super bimagic total labeling. Secondly, the idea of super edge trimagic total graph labeling was introduced. We found some complicated graphs with trimagic total numbers, including the triangulated wheel graph, the double vertex wheel graph, and the closed triangulated water wheel graph.