Tur\'an Graphs, Stability Number, and Fibonacci Index

TL;DR

This paper establishes tight upper bounds for the Fibonacci index of graphs based on stability number and order, highlighting Turán graphs as extremal structures in these bounds.

Contribution

It introduces new extremal bounds for the Fibonacci index related to stability number and demonstrates Turán graphs' extremality in this context.

Findings

Turán graphs are extremal for Fibonacci index bounds.

Tight upper bounds are established for general and connected graphs.

Connected variants of Turán graphs are also extremal.

Abstract

The Fibonacci index of a graph is the number of its stable sets. This parameter is widely studied and has applications in chemical graph theory. In this paper, we establish tight upper bounds for the Fibonacci index in terms of the stability number and the order of general graphs and connected graphs. Tur\'an graphs frequently appear in extremal graph theory. We show that Tur\'an graphs and a connected variant of them are also extremal for these particular problems.