Fibonacci Index and Stability Number of Graphs: a Polyhedral Study

TL;DR

This paper investigates the Fibonacci index of graphs, establishing tight bounds related to stability number and graph order, and characterizes extremal graphs using polyhedral methods.

Contribution

It provides tight upper bounds for the Fibonacci index in relation to stability number and graph order, and characterizes extremal graphs through polyhedral analysis.

Findings

Turán graphs are extremal for Fibonacci index bounds.

All optimal linear inequalities for stability number and Fibonacci index are characterized.

Polyhedral study reveals the structure of extremal graphs.

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. We also make a polyhedral study by establishing all the optimal linear inequalities for the stability number and the Fibonacci index, inside the classes of general and connected graphs of order $n$.