A Unified Framework for Degree-Based Topological Indices in Distance-Hereditary Fuzzy Graphs

Abstract

Distance-Hereditary Fuzzy Graphs or DHFGs combine the distance preserving property of distance- graphs with the ability of fuzzy graphs to show uncertainty. Many degree-based topological indices have been studied for fuzzy graphs. A unified framework for these indices in DHFGs has not been created yet. In this paper we suggest a framework for creating degree-based topological indices in Distance-Hereditary Fuzzy Graphs. This framework includes known indices, as special cases. These are the Randic, Zagreb, Harmonic, Atom-Bond Connectivity or ABC, Geometric-Arithmetic or GA and Sombor indices. We prove fundamental properties such as existence, non-negativity, monotonicity, invariance under fuzzy isomorphisms, and preservation under connected induced distance-hereditary fuzzy subgraphs. Illustrative examples for the computation of the proposed indices are given, and their practical relevance is illustrated by applications to communication networks, transportation systems, biological interaction networks, social networks and decision-support systems. The proposed unified framework provides a systematic basis for extending classical topological descriptors to uncertain distance-preserving networks, and also opens new avenues for future research in the theory of fuzzy graphs.