In this study, we introduce the concept of the K-Lexicographic Max Product (K−LMP) of neutrosophic graphs and explore its associated degree structure to enhance decision-making frameworks in food safety applications related to risk assessment, including freshness, contamination, and spoilage. Neutrosophic graphs, capable of handling indeterminacy, inconsistency, and incompleteness, provide a flexible mathematical foundation for modelling complex systems. By incorporating the K−LMP into neutrosophic graphs, we offer a novel approach to comparing and ranking food safety scenarios where multiple attributes and uncertain information coexist. We present example graphs and theorems related to K−LMP and further define the K-Lexicographic degree to quantify node significance within the context of neutrosophic graphs. To validate the practical utility of this approach, a food safety analysis is implemented, demonstrating how the model identifies critical control points and supports more robust, transparent decision-making under uncertainty. This work contributes to the advancement of neutrosophic graph theory and its interdisciplinary application in food quality and safety management.

In applications requiring uncertain, imprecise, and multi-dimensional data, where traditional distance measures frequently fall short of capturing the full complexity of interactions among elements, a distance measure for complex intuitionistic fuzzy sets (DMCIFSs) becomes essential. Although DMCIFSs have been developed, most of them do not account for the hesitation degree, which is crucial for capturing ambiguity and uncertainty in human reasoning. As extensions of the normalized Hamming and Euclidean distance measures, this work proposes two new measures namely the Hesitance DMCIFSs (HDMCIFSs) and the Euclidean Hesitance DMCIFSs (EHDMCIFSs). These newly proposed measures provide a more comprehensive framework for modeling uncertainty by explicitly incorporating the hesitancy component. In addition to the proposed measures, several fundamental procedures and theoretical results are also presented. Furthermore, a novel decision-making method utilizing these distance measures is developed and applied to multi-criteria decision-making (MCDM) problems. The effectiveness of the proposed methods is demonstrated through a comparative study, highlighting their potential for improved sensitivity and accuracy in practical decision-making scenarios.