First Zagreb index on a fuzzy graph and its application
Sk Rabiul Islam, Madhumangal Pal
Abstract
The Zagreb index (ZI) is a very important graph parameter and it is extensively used in molecular chemistry, spectral graph theory, network theory and several fields of mathematics and chemistry. In this article, the first ZI is studied for several fuzzy graphs like path, cycle, star, fuzzy subgraph, etc. and presented an ample number of results. Also, it is established that the complete fuzzy graph has maximal first ZI among n-vertex fuzzy graphs. Some bounds of first ZI are discussed for Cartesian product, composition, union and join of two fuzzy graphs. An algorithm has been designed to calculate the first ZI of a fuzzy graph. At the end of the article, a multi-criteria decision making (MCDM) method is provided using the first ZI of a fuzzy graph to find the best employee in a company. Also a comparison is provided among related indices on the result of application and shown that our method gives better results.