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The edge-isoperimetric number of graphs and their powers: approaches from spectral graph theory, optimization and finite geometry

Aida Abiad, Nils van de Berg, Emanuel Juliano, Harper Reijnders, Robin Simoens, Thijs van Veluw, Jim Wittebol

Abstract

We obtain several sharp spectral bounds, approximations, and exact values for the isoperimetric number and related edge-expansion parameters of graphs. Our results focus on graph powers and on families of graphs with rich algebraic or geometric structure, including distance-regular graphs and graphs arising from finite geometries, among others. Our proofs use techniques from spectral graph theory, linear optimization, finite geometry, and probability, yielding new machinery for analysing edge-expansion phenomena in highly structured graphs.

The edge-isoperimetric number of graphs and their powers: approaches from spectral graph theory, optimization and finite geometry

Abstract

We obtain several sharp spectral bounds, approximations, and exact values for the isoperimetric number and related edge-expansion parameters of graphs. Our results focus on graph powers and on families of graphs with rich algebraic or geometric structure, including distance-regular graphs and graphs arising from finite geometries, among others. Our proofs use techniques from spectral graph theory, linear optimization, finite geometry, and probability, yielding new machinery for analysing edge-expansion phenomena in highly structured graphs.
Paper Structure (18 sections, 27 theorems, 71 equations, 1 figure, 6 tables)

This paper contains 18 sections, 27 theorems, 71 equations, 1 figure, 6 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Eigenvalue bounds for the isoperimetric number of graph powers
  4. Bounds when the spectra of $G$ and $G^t$ are related
  5. Bounds when the spectra of $G$ and $G^t$ are not related
  6. Optimal polynomials in Theorem \ref{['theorem:lowerboundi(G^t)']} and Theorem \ref{['theorem:upperboundi(G^t)']} when $t=2$
  7. Optimization of the bounds in Theorem \ref{['theorem:lowerboundi(G^t)']} and Theorem \ref{['theorem:upperboundi(G^t)']}
  8. Approximating the isoperimetric number of distance-regular graphs
  9. The exact isoperimetric number and a connection with finite geometry
  10. Graph classes whose isoperimetric number is known
  11. Hamming graphs and other Cartesian products
  12. Johnson graphs
  13. Tight sets
  14. Grassmann graphs
  15. Polar graphs
  16. ...and 3 more sections

Key Result

Theorem 2.1

Let $G$ be a $k$-regular graph on $n$ vertices with Laplacian eigenvalues $0=\mu_1\le\dots\le\mu_n$. For any subset $S$ of the vertices we have If equality holds in either inequality, then $S$ is an intriguing set.

Figures (1)

  • Figure 1: $S_m=S_{m_0}\cup T$, where $T$ is an initial segment of the "slice" $H$.

Theorems & Definitions (53)

  • Theorem 2.1: Haemers1995
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Theorem 3.4
  • proof
  • Theorem 3.5
  • proof
  • Remark 3.6
  • ...and 43 more