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

TL;DR

This paper develops new spectral, optimization, and geometric methods to analyze the edge-isoperimetric number and edge-expansion properties of graphs, especially those with algebraic or geometric structures like distance-regular graphs.

Contribution

It introduces novel bounds, approximations, and exact values for edge-expansion parameters using interdisciplinary techniques from spectral graph theory, finite geometry, and optimization.

Findings

Sharp spectral bounds for isoperimetric numbers

Exact values for edge-expansion in structured graphs

New analytical tools for highly regular graphs

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.