Topology Induced by Fuzzy Coloring in Fuzzy Graphs

In fuzzy graphs, the concept of fuzzy topology induced by fuzzy coloring on the vertex set of a fuzzy graph is a relatively advanced and specialized topic that combines the principles of fuzziness, graph theory, and topology. Fuzzy topology typically refers to the study of topological structures where the open sets are not crisp but fuzzy in nature, meaning they have degrees of membership rather than being either fully included or excluded. When fuzzy coloring is applied to set of vertices of a fuzzy graph, it leads to the creation of fuzzy topological spaces where the relationships between the vertices are characterized by fuzzy sets or fuzzy relations. This interplay between fuzzy coloring and fuzzy topology can help model systems with imprecise, uncertain, or gradated connectivity. This chapter provides the technique of inducing fuzzy topology on fuzzy graph’s set of vertices induced by vertex coloring. It initiates color lower fuzzy approximation and color upper fuzzy approximation of fuzzy subgraphs and defines the closed and open sets of the fuzzy topology generated by fuzzy chromatic partition. It explores key properties of color lower fuzzy approximation and color upper fuzzy approximation of fuzzy subgraphs. It also establishes some real-time applications of the color lower fuzzy approximation and color upper fuzzy approximation.