Advancing Uncertain Combinatorics through Graphization, Hyperization, and Uncertainization: Fuzzy, Neutrosophic, Soft, Rough, and Beyond (2411.17411v1)

Abstract: To better handle real-world uncertainty, concepts such as fuzzy sets, neutrosophic sets, rough sets, and soft sets have been introduced. For example, neutrosophic sets, which simultaneously represent truth, indeterminacy, and falsehood, have proven to be valuable tools for modeling uncertainty in complex systems. These set concepts are increasingly studied in graphized forms, and generalized graph concepts now encompass well-known structures such as hypergraphs and superhypergraphs. Furthermore, hyperconcepts and superhyperconcepts are being actively researched in areas beyond graph theory. Combinatorics, uncertain sets (including fuzzy sets, neutrosophic sets, rough sets, soft sets, and plithogenic sets), uncertain graphs, and hyper and superhyper concepts are active areas of research with significant mathematical and practical implications. Recognizing their importance, this paper explores new graph and set concepts, as well as hyper and superhyper concepts, as detailed in the "Results" section of "The Structure of the Paper." Additionally, this work aims to consolidate recent findings, providing a survey-like resource to inform and engage readers. For instance, we extend several graph concepts by introducing Neutrosophic Oversets, Neutrosophic Undersets, Neutrosophic Offsets, and the Nonstandard Real Set. This paper defines a variety of concepts with the goal of inspiring new ideas and serving as a valuable resource for researchers in their academic pursuits.