**Thesis Type:** Postgraduate

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

**Approval Date:** 2016

**Thesis Language:** Turkish

**Student:** MERVE AŞÇIOĞLU

**Supervisor: **İSMAİL NACİ CANGÜL

The aim of this work is to make some calculations with some special graph classes by stating a mathematical invariant used in chemical applications by several formulae and using graph theoretical methods, to obtain relations between this invariant and possible graphs and to find relations for Narumi-Katayama index of the join operation on graphs. This thesis consists of four chapters. The first chapter is introduction. Here the definition of a graph, its history, some spacial graphs and their properties together with some new graph types which do not exist in literature. These information will be used throughout the thesis. In the second chapter, the Narumi-Katayama index is calculated for some well-known special graph classes and a generalisation is given. In the third chapter, the Narumi-Katayama index of the join of two graphs is calculated and some results are obtained regarding a theorem given by Azari, [Sharp lower bounds on the Narumi-Katayama index of graph operations, 2014]. In the fourth chapter, the number of graphs of which the Narumi-Katayama index is equal to a given fixed number.