Colouring in fuzzy graphs and inverse fuzzy graphs algorithms applications and studies on chess puzzles

Abstract

This thesis, titled "Colouring in Fuzzy Graphs and Inverse Fuzzy Graphs: Algorithms, Applications, andStudies on Chess Puzzles," presents a comprehensive study on the application and extension of fuzzygraphs and inverse fuzzy graphs in solving real-world problems, focusing colouring, primarily ondisease transmission path from a set of countries to another set, decision-making in chess puzzles andsimultaneous drug intake analysis. The research includes some novel fuzzy graph operations,innovative colouring techniques, and new theoretical constructs like the chromatic number andchromatic polynomial for inverse fuzzy graphs.Initially, a fuzzy graph-based model is developed to analyse the transmission pathways ofCOVID-19. By clustering a set of countries using a g-distance metric and constructing a fuzzy graph byconsidering clusters of a particular order as vertices, the model identifies the most dangerous minimalpaths between clusters. This approach effectively highlights the nations with high infection rates andreveals the minimal yet most threatening paths of viral spread.The study further extends to operations on fuzzy graphs by constructing new operations such asthe u-product, strong edge product, and k th power. The relationship between the fuzzy chromaticnumber of these resultant graphs and their component graphs is examined using fuzzy colouring.The thesis then goes into a novel application, by modelling a chess puzzle as a fuzzy graph andusing colouring technique of fuzzy graph, determine the number of possible captures within thepuzzle. Later the study extends to identifying better moves in critical chess scenarios.To address the limitations in existing fuzzy graph theory, the researchers introduces a newconcept called the inverse fuzzy graph ( I-fuzzy graph). This study focused on defining colouring in anI-fuzzy graph. Chromatic number of an I-fuzzy graph is defined and analysed under operations like I-fuzzy union, intersection, and restricted union. This Inverse fuzzy graph colouring study shows how itcan be used in real life, such as modelling drug interactions.Finally, the thesis defines the chromatic polynomial of an inverse fuzzy graph using inverse fuzzygraph colouring based on MMV edges. This work identifies the characteristics of the inverse fuzzychromatic polynomial. Through these contributions, the thesis not only enhances the theoreticalfoundations of fuzzy graph colouring but also demonstrates its diverse applications in epidemiology,artificial intelligence, and healthcare decision-making.