Centrality–Accessibility Sombor Index for Fuzzy Graphs

Abstract

A graph index under fuzzy uncertainty should respond to both uncertain edges and the structural roles of their endpoints. A centrality-accessibility Sombor index is introduced by combining normalized fuzzy degree with max-min network accessibility at each vertex and then aggregating edge contributions through a Sombor-type Euclidean norm. The construction reduces to a scaled classical Sombor index on crisp graphs, is invariant under fuzzy-graph isomorphism, and admits universal bounds. Monotonicity with respect to the local-global preference parameter is characterized, while a perturbation theorem controls index variation when edge grades or centrality values change. A six-node transmission network is calculated edge by edge: the balanced index equals 4.4657, with the RS relation contributing most. The index decreases from 5.8056 to 3.5086 as emphasis moves from accessibility to local degree, exposing the distinction between a globally coherent and locally concentrated fuzzy network. Edge-contribution profiles and uncertainty ranges permit network comparison while retaining the individual relations responsible for the index value.