Embedding of Supplementary Results in Strong EMT Valuations and Strength

A graph ℘ is said to be edge-magic total (EMT if there is a bijection Υ : V(℘) ∪ E(℘) → {1, 2, …, |V(℘) ∪ E(℘)|} s.t., Υ(υ) + Υ(υν) + Υ(ν) is a constant for every edge υν ∈ E(℘). An EMT graph ℘ will be called strong edge-magic total (SEMT) if Υ(V(℘)) = {1, 2, …, |V(℘)|}. The SEMT strength, sm(℘), of a graph ℘ is the minimum of all magic constants a(Υ), where the minimum runs over all the SEMT valuations of ℘, this minimum is defined only if the graph has at least one such SEMT valuation. Furthermore, the SEMT deficiency of a graph ℘, μs(℘), is either the minimum non-negative integer n such that ℘ ∪ nK1 is SEMT or +∞ if there will be no such integer n. In this paper, we will present the strong edge-magicness and deficiency of disjoint union of 2-sided generalized comb with bistar, path and caterpillar, moreover we will evaluate the SEMT strength for 2-sided generalized comb.