The structural properties of fuzzy median graphs

This paper investigates the relationship between fuzzy median graphs and blocks, which are important concepts in fuzzy graph theory, particularly in terms of connectivity. The main focus is on the fuzzy median property of a block that has at least four vertices and does not have any fuzzy bridges. The paper also examines the fuzzy median property of isomorphic images of median graphs and explores the median behavior of isometric subgraphs of a fuzzy median graph. Furthermore, the paper investigates the median property of fuzzy median graphs under graph operations. The main concern of the paper is to determine whether the union of two fuzzy median graphs retains fuzzy median behavior. The paper studies the connection between fully strong gated fuzzy graphs and fuzzy median graphs. It also explores whether the gate of an induced subgraph can also be in the median set of any triple of vertices. Additionally, the paper introduces a novel centrality measure called median centrality to identify the influential nodes in a network. The sum of the median centrality of all vertices in a fuzzy median graph will always be one.