arXiv:2605.23936v1 Announce Type: new Abstract: This book presents a comprehensive and systematic survey of graph theory under uncertainty, with particular emphasis on the unifying role of the uncertain graph framework. It reviews fundamental concepts, structural properties, graph classes, and graph parameters within fuzzy, neutrosophic, and related models, while also introducing a wide range of extensions such as uncertain digraphs, hypergraphs, superhypergraphs, and dynamic graphs. In addition to theoretical developments, the book explores practical applications, including uncertain molecular graphs, decision-making systems, graph neural networks, knowledge graphs, and cognitive maps. By organizing diverse uncertainty-aware graph models within a common perspective, this work provides a coherent framework for understanding their relationships, capabilities, and applications in complex systems.

arXiv:2605.24053v1 Announce Type: new Abstract: Large Language Models (LLMs) are predominantly governed by probabilistic frameworks in which the sum of outcome probabilities is constrained to unity. This architectural limitation, often imposed by Softmax layers, leads to a collapse of uncertainty that makes it difficult to differentiate between epistemic uncertainty, paradox, and vagueness. We present an empirical investigation of the application of Neutrosophic Logic, a framework that treats Truth (T), Indeterminacy (I), and Falsity (F) as three independent dimensions, to model epistemic states in LLMs. We conducted experiments on a family of four OpenAI GPT models across five linguistic phenomena: logical paradoxes, epistemic ignorance, vagueness, ethical contradictions, and future contingencies, under three prompting strategies: neutrosophic, probabilistic, and entropy-derived. Our findings reveal that the neutrosophic approach, by allowing T+I+F > 1, a state we term hyper-truth, provides a richer representation of a model's internal state. In 35% of evaluations, hyper-truth emerged spontaneously, predominantly under ethical contradiction and logical paradox. We demonstrate that this approach preserves truth values in fuzzy contexts and offers a robust method for identifying and quantifying internal model conflict. We conclude that the integration of neutrosophic evaluation layers is a critical step toward more transparent, reliable, and ethically aware AI systems.

arXiv:2411.17411v2 Announce Type: replace Abstract: Combinatorics studies how discrete objects can be counted, arranged, and combined under specified rules. Motivated by uncertainty in real-world data and decisions, modern set-theoretic formalisms such as fuzzy sets, neutrosophic sets, rough sets, soft sets, and plithogenic sets have been developed. In particular, neutrosophic sets model uncertainty by assigning to each element degrees of truth, indeterminacy, and falsity. In parallel, these uncertainty frameworks are increasingly investigated in graphized and hyperized forms, where generalized graph models encompass classical graphs, hypergraphs, and higher-order "superhyper" structures; related hyper- and superhyper-concepts also arise beyond graph theory. This book (Edition 2.0) surveys and consolidates recent developments at the intersection of combinatorics, uncertain sets, uncertain graphs, and hyper/superhyper frameworks, while introducing several new graph and set concepts. As representative contributions, we extend graph-theoretic notions via Neutrosophic Oversets, Neutrosophic Undersets, Neutrosophic Offsets, and the Nonstandard Real Set. The second edition adds newly introduced concepts, corrects typographical issues, and re-examines mathematical consistency, aiming to serve as a compact reference and a source of inspiration for further research.