Abstarct: In this thesis we investigate the fuzzy molecular graphs. In the first chapter some definitions and preliminaries are presented. The second chapter is dedicated to introducing some topological indices related to the molecular graph. In the chapter third we define fuzzy molecular graph by using the concept of fuzzy numbers. The fourth chapter is devoted to calculation of some fuzzy topological indices baxsed on fuzzy molecular graph. keywords: Molecular graph, number fuzzy, topological indices