Sombor index of fuzzy graph

Abstract

Topological indices (TI) play a crucial role across various research domains, including network theory, spectral graph theory, and molecular chemistry. These indices are created primarily in the context of crisp graphs, but they can also be applied to fuzzy graphs, which are a more generalized version of crisp graphs. This article presents the Sombor index for fuzzy graphs (SOF(G)) and explores how it can be applied to diverse categories of fuzzy graphs, such as cycles, stars, complete graphs, and fuzzy subgraphs. Results are obtained by this index after vertices and edges are eliminated and we also proved that it holds for isomorphic fuzzy graphs and established interesting bounds for SOF(G). Along with several theorems and examples, the Sombor index is introduced and explored for fuzzy directed graphs (FDG), regular fuzzy graphs (RFGs), and fuzzy cycles. Furthermore, a connection between the Sombor index and other fuzzy graph indices is established. Finally, an application is provided demonstrating the use of the Sombor index of a fuzzy graph to identify the country with the optimal case of human trafficking.