Hub Number in Ring Fuzzy Graphs and Restricted Hyper-Totient Fuzzy Graphs

Hub Number in Ring Fuzzy Graphs

Özet

In this paper, we introduce and study the hub number of ring fuzzy graphs. To make the construction self-contained, a ring fuzzy graph is treated as a weighted fuzzy structure RFΓ = (V, E, μ, σ) whose underlying vertices are ring-induced substructures and whose hub number is computed by the total vertex-membership value of a minimal hub set. We investigate this parameter for several important classes of ring fuzzy graphs, including complete, path, cycle, and star ring fuzzy graphs, and obtain explicit expressions in each case. In addition, we derive general bounds for the hub number in terms of the complement graph and several neighborhood-based parameters. Relationships between the hub number and other graph-theoretic quantities, such as domination-type and chromatic-type parameters, are also examined. Furthermore, we extend the study to restricted hyper-totient fuzzy graphs and present a characterization of their hub number. Finally, we analyze the behavior of the hub number under several operations on ring fuzzy graphs, including intersection, union, direct product, and direct sum, stating the required compatibility assumptions on the underlying vertex sets and membership functions. The results obtained in this work show that the hub number is a useful structural parameter for the analysis of ring fuzzy graphs and provide a foundation for further investigations in algebraic fuzzy graph theory.