Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Fibonacci Index and Stability Number of Graphs: a Polyhedral Study

Véronique Bruyère, Hadrien Mélot

TL;DR

This paper establishes tight upper bounds for the Fibonacci index in terms of the stability number and the order of general graphs and connected graphs and shows that Turán graphs and a connected variant of them are also extremal for these particular problems.

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. Turán graphs frequently appear in extremal graph theory. We show that Turán 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 $n$.

Fibonacci Index and Stability Number of Graphs: a Polyhedral Study

TL;DR

This paper establishes tight upper bounds for the Fibonacci index in terms of the stability number and the order of general graphs and connected graphs and shows that Turán graphs and a connected variant of them are also extremal for these particular problems.

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. Turán graphs frequently appear in extremal graph theory. We show that Turán 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 .

Paper Structure

This paper contains 11 sections, 21 theorems, 37 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Basic properties
  3. Notations
  4. $\alpha$-critical graphs
  5. Fibonacci index
  6. General graphs
  7. Connected graphs
  8. 1) $G$ has an $\alpha$-critical decomposition.
  9. 2) $G$ is $\alpha$-critical.
  10. Polyhedral study
  11. Observations

Key Result

Lemma 1

Let $G$ be an $\alpha$-critical graph. If $G$ is connected, then the graph $G^{v}$ is connected for all vertices $v$ of $G$.

Figures (5)

  • Figure 1: The graphs ${\sf CS}_{7,3}$, ${\sf T}_{7,3}$ and ${\sf TC}_{7,3}$
  • Figure 2: Cliques in the graph $G$
  • Figure 3: The polytopes ${\mathcal{P}}_{{\cal G}(10)}$ (left) and ${\mathcal{P}}_{{\cal C}(10)}$ (right)
  • Figure 4: Representation of ${\mathcal{P}}_{{\cal G}(n)}$ and ${\mathcal{P}}_{{\cal C}(n)}$ together
  • Figure 5: Graphs with same order and stability number

Theorems & Definitions (42)

  • Definition 1
  • Example 1
  • Example 2
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Lemma 3
  • Example 3
  • Lemma 4
  • ...and 32 more