Extended Fuzzy H-Magic Labelings of Some Graphs

Abstract

Let G⋆ = (V,E) and H = (V⋆,E⋆) be simple, finite, planar, connected and undirected graphs, where every edge of G⋆ belongs to at least one subgraph of G⋆ isomorphic to H. A fuzzy labeling graph G = (κ,τ) on G⋆ is defined by the mappings κ : V → [0,1] and τ : E → [0,1], which are one-to-one and satisfy the condition: τ(uv)<min(κ(u),κ(v)) ∀ uv ∈ E. A fuzzy labeling graph G = (κ,τ) on G⋆ is called an Intuitionistic fuzzy H-magic labeling graph if there exist constants m1 , m2 ∈ (0,v⋆+e⋆) such that for any subgraph H⋆ of G isomorphic to H, the following conditions hold: Σu∈V⋆κP(u)+Σv∈E⋆κP(v) = m1 Σu∈V⋆κN(u)+Σv∈E⋆κN(v) = m2 Further, a Neutrosophic Fuzzy H-Magic Labeling Graph is defined if there exist constants m1 , m2 and m3 ∈ (0,v⋆+e⋆), representing positive grade, indeterminacy, and negative grade, respectively, such that: Σu∈V⋆κP(u)+Σv∈E⋆κP(u) = m1 Σu∈V⋆κI(u)+Σv∈E⋆κI(v) = m2 Σu∈V⋆κN(u)+Σv∈E⋆κN(v) = m3 In this thesis, I have created and studied new methods for labeling specific types of graphs such as book graphs, generalized prism graphs and ladder graphs using intuitionistic fuzzy and neutrosophic fuzzy H-magic labelings of graphs. In areas like network analysis and decision-making, this study facilitates understanding of graph uncertainty management.