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On Expansion of Random Regular Graphs: Improved Lower Bounds for Small Even Degrees

Pasin Manurangsi

Abstract

We show that a simple scoring-based tie-breaking can help improve lower bounds for the expansion (aka isoperimetric number) of random regular graphs with small even degrees. Specifically, for degrees 4, 6 and 8, we show that, with high probability, the expansions are at least 0.489, 1.120 and 1.813 respectively.

On Expansion of Random Regular Graphs: Improved Lower Bounds for Small Even Degrees

Abstract

We show that a simple scoring-based tie-breaking can help improve lower bounds for the expansion (aka isoperimetric number) of random regular graphs with small even degrees. Specifically, for degrees 4, 6 and 8, we show that, with high probability, the expansions are at least 0.489, 1.120 and 1.813 respectively.

Paper Structure

This paper contains 19 sections, 9 theorems, 44 equations, 1 table.

Table of Contents

  1. Introduction
  2. Our Contributions.
  3. Overview of Techniques.
  4. Other Related Work.
  5. Preliminaries
  6. Random Regular Graphs and Expansion
  7. Configuration Vector
  8. Stirling's Approximation
  9. Minimizing Relative Entropy Subject to Linear Constraints
  10. Tie-Breaking Violating Sets
  11. Local Improvement
  12. Scoring-Based Tie-Breaking
  13. Improved Bound for Expansion: Proof of \ref{['thm:main']}
  14. Symmetric Case: $\alpha = 1/2$
  15. Case 1: $\Delta = 4$.
  16. ...and 4 more sections

Key Result

Theorem 1

Let $\Delta \in \{4, 6, 8\}$, and let $\nu^*_{\Delta}$ be such that $H_{\Delta}(\nu^*_{\Delta}/2) = 0$ where $H_{\Delta}$ is as defined in eq:def-for-main-bound. In particular, $\nu^*_4 = 0.4894\dots, \nu^*_6 = 1.1205\dots, \nu^*_8 = 1.8130\dots$. Then, for any constant $\nu < \nu^*_{\Delta}$, a gra

Theorems & Definitions (15)

  • Theorem 1
  • Theorem 2: Bollobas88
  • Theorem 3: KolesnikW14
  • Lemma 1: Lampis12
  • Theorem 4
  • proof
  • Lemma 2: Local Optimality
  • proof
  • Lemma 3
  • Lemma 4
  • ...and 5 more