Embedding of Neutrosophic Graphs on Topological Surfaces (JOURNAL OF UNIVERSITY OF BABYLON for Pure and Applied Sciences, 2024)
Background: A planar graph (PG) is a graph with no intersecting edges. Particular to both crisp and neutrosophic graphs (NG) is the planar graph, in contrast to crisp planar graphs. NPGs allow for the intersection of neutrosophic edge NEs, since the value of planarity in these graphs is the degree of planarity of the intersected NEs. The NPGs are often represented on a flat surface.Materials and Methods: This study discusses how to embed NGs on surfaces such as spheres and m-toruses by defining the degree of intersection of the neutrosophic edges of NGs with finding the faces on the given graph structures using Euler's theorems. Here, the proofs of Euler's theorems help us find, given the total NFV of G, the interval containing that value.Result: As result of this work obtained that for any two isomorphic planer graphs, they have the same planarity value. For any neutrosophic planer graph with f = (1,1,1) can be embedded in the plane if it can be embedded in the sphere and according to NPGs, for planar and spherical surfaces, equivalent theorems to Euler's formula are proved and shown.Conclusion: It concludes that by using neutrosophic sets and crisp graphs to construct neutrosophic graphs with the benefit of Euler’s theorem, it can provide the concept of embedding neutrosophic graphs in different topological surfaces such as a plane, sphere, and m-torus.