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Analogues of Bermond-Bollobás Conjecture for Cages Yield Expander Families

Leonard Chidiebere Eze, Robert Jajcay

Abstract

This paper presents a possible link between Cages and Expander Graphs by introducing three interconnected variants of the Bermond and Bollobás Conjecture, originally formulated in 1981 within the context of the Degree/Diameter Problem. We adapt these conjectures to cages, with the most robust variant posed as follows: Does there exist a constant $c$ such that for every pair of parameters $(k,g)$ there exists a $k$-regular graph of girth $g$ and order not exceeding $ M(k,g) + c $?; where $M(k,g)$ denotes the value of the so-called Moore bound for cages. We show that a positive answer to any of the three variants of the Bermond and Bollobás Conjecture for cages considered in our paper would yield expander graphs (expander families); thereby establishing a connection between Cages and Expander Graphs.

Analogues of Bermond-Bollobás Conjecture for Cages Yield Expander Families

Abstract

This paper presents a possible link between Cages and Expander Graphs by introducing three interconnected variants of the Bermond and Bollobás Conjecture, originally formulated in 1981 within the context of the Degree/Diameter Problem. We adapt these conjectures to cages, with the most robust variant posed as follows: Does there exist a constant such that for every pair of parameters there exists a -regular graph of girth and order not exceeding ?; where denotes the value of the so-called Moore bound for cages. We show that a positive answer to any of the three variants of the Bermond and Bollobás Conjecture for cages considered in our paper would yield expander graphs (expander families); thereby establishing a connection between Cages and Expander Graphs.
Paper Structure (8 sections, 9 theorems, 27 equations)

This paper contains 8 sections, 9 theorems, 27 equations.

Table of Contents

  1. Introduction
  2. Variants of Bermond-Bollobás Conjecture for Cages
  3. Odd Girth
  4. Even Girth
  5. Concluding Remarks
  6. Acknowledgments
  7. Declaration of competing interest
  8. Data availability

Key Result

Theorem 1

Let $\Gamma$ be a $k$-regular connected graph, then where $\lambda$ is the second largest eigenvalue of $\Gamma$.

Theorems & Definitions (16)

  • Theorem 1: Alon
  • Theorem 2: B
  • Lemma 3
  • proof
  • Lemma 4
  • proof
  • Theorem 5
  • proof
  • Lemma 6
  • proof
  • ...and 6 more