(untitled) (JOURNAL OF UNIVERSITY OF BABYLON for Pure and Applied Sciences, 2024)
Background: Although graph theory plays a significant role in representing the causes of a problem and the relationship between them to facilitate its solution, there are many problems in our real lives that cannot be represented accurately due to the inaccuracy and ambiguity of the available data. In this article, Cayley (SVNG) will define and look into some of its characteristics. In terms of algebraic structures, we demonstrate a few intriguing characteristics of SVNGs. Additionally, planarity in Cayley (SVNG)s will be discussed. Materials and Methods: The single-valued Neutrosophic sets (SVNS) were used, which depend on three functions from the universal set, say X, to the standard range [0, 1] and emanating from the non-standard range ]-0, [ which the Neutrosophic sets rely on. The Cayley table of algebraic groups was used. Result: This article introduced the idea of neutrosophic Cayley graphs (NCG) as a combination of graph theory and algebraic structure, and some of its properties were studied in different environments where the algebraic Cayley structure exists. For example, it was proven that every reflexive and transitive Cayley graph is regular and so on. Conclusion: It can be concluded that a relationship has been built to draw a weighted directed graph for SVNS that contains three components (truth, indeterminacy, and false). and many interesting features of the neutrosophic graph (NG) were displayed, such as transitivity and regularity. The planarity of Cayley’s neutrosophic graph has also been demonstrated, and observations and theorems have been formulated in relation to it.