F-Index of Some Graph Operations

TL;DR

This paper investigates how the F-index, a topological graph invariant based on vertex degrees, behaves under various graph operations and applies these findings to molecular and nanostructure graphs.

Contribution

It provides new insights into the behavior of the F-index under multiple graph operations and computes its values for various molecular and nanostructure graphs.

Findings

F-index behavior characterized for several graph operations.

Formulas derived for F-index of molecular graphs.

Applications demonstrated on nanostructure graphs.

Abstract

The F-index of a graph is defined as the sum of cubes of the vertex degrees of the graph. This was introduced in 1972, in the same paper where the first and second Zagreb indices were introduced to study the structure-dependency of total $\pi$-electron energy. But this topological index was not further studied till then. Very recently, Furtula and Gutman [B. Furtula, I. Gutman, A forgotten topological index, J. Math. Chem., 53(4)(2015) 1184--1190.] reinvestigated the index and named it "forgotten topological index" or "F-index". In that paper, they present some basic properties of this index and showed that this index can enhance the physico-chemical applicability of Zagreb index. Here, we study the behavior of this index under several graph operations and apply our results to find the F-index of different chemically interesting molecular graphs and nano-structures.