Edge-based first Zagreb index for fuzzy graphs

Abstract

Topological indices have gained much importance in fuzzy graph (FG) theory and provide valuable information regarding the structure of graphs. This paper provides an innovative edge-based version of the first Zagreb index (ZI) for FGs. The proposed ZI on FG is comprehensively studied in case of different classes of FGs, such as fuzzy path, fuzzy cycle, and fuzzy star graph. It is also discussed how this index's behaves when the FG is isomorphic with one of its fuzzy subgraphs. Moreover, some sharp bounds on the proposed index for various operations on FGs are derived. It should be mentioned that the presented results not only contribute to the general theory of FGs but are also useful for various practical purposes, such as determining properties of molecular graphs in molecular chemistry.