**Thesis Type:** Doctorate

**Institution Of The Thesis:** Uludağ Üniversitesi, Turkey

**Approval Date:** 2014

**Thesis Language:** Turkish

**Student:** MÜGE TOGAN

**Supervisor: **İSMAİL NACİ CANGÜL

In this work, subdivision graphs are recalled, r-subdivision graphs are defined and ten types of Zagreb indices of these graphs are calculated. This application shows that it is enough to know only the number of vertices and edges of the graphs, instead of dealing with the degrees of all vertices of the graphs and it provides great convenience for the calculation of the Zagreb indices. This thesis consists of four chapters. First chapter is introduction, and a brief summary of related literature and the necessary preliminaries are given in this chapter. Some basic concepts which will be used in the forthcoming chapters are introduced here. In the second chapter, Zagreb and multiplicative Zagreb indices and coindices of graphs, total multpilicative sum Zagreb index and multpilicative sum Zagreb index are introduced and some results and theorems for all these Zagreb indices are given. In the third chapter, ten types of Zagreb indices are calculated for some well-known graphs, such as path graph, cycle graph, star graph, complete graph, complete bipartite graph and tadpole graph and some results are obtained. In the fourth chapter, ten types of Zagreb indices of subdivision and r-subdivision graphs for some well-known graphs are given and some inequalities which shows the relations between several Zagreb indices of subdivision graphs are obtained.