Second Zagreb index for fuzzy graphs and its application in mathematical chemistry

The Zagreb index (ZI) of a crisp graph and also for a fuzzy graph (FG) is a very much useful tool in network theory, spectral graph theory, molecular chemistry and several fields of chemistry and mathematics. The second ZI is studied for FGs here. Bounds of this index are calculated for several FGs: path, star, cycle, complete FG, partial fuzzy subgraph, etc. For isomorphic FGs, it is shown that the value of this index is same. Bounds of this index for the Cartesian product, composition, join and union of two FGs are established. At the end of this article, an application of the index in mathematical chemistry is studied. For this, octane isomers are considered and analyzed the correlation between this index with some physico-chemical properties of octane isomers. This index's correlation coefficient (r) with acentric factor and entropy is determined for the linear curve fittings. Using the value of r, one can conclude that this index can help estimate the acentric factor and entropy with significant accuracy. Also, these outcomes declare the  appropriateness of the index in QSPR research.

Keywords:

Fuzzy graph , second Zagreb index , topological index , chemical graph