The Boundary of a Graph and its Isoperimetric Inequality

TL;DR

This paper introduces a new boundary concept for graphs that aligns with Euclidean domain discretizations and establishes an isoperimetric inequality linking boundary size, number of vertices, and diameter.

Contribution

It defines a boundary for graphs consistent with Euclidean boundaries and proves an isoperimetric inequality relating boundary size, graph order, and diameter.

Findings

The boundary coincides with Euclidean domain boundaries for discretizations.

Graphs with many vertices have large boundaries unless they contain long paths.

The inequality scales similarly to classical Euclidean isoperimetric principles.

Abstract

We define, for any graph $G=(V,E)$, a boundary $\partial G \subseteq V$. The definition coincides with what one would expected for the discretization of (sufficiently nice) Euclidean domains and contains all vertices from the Chartrand-Erwin-Johns-Zhang boundary. Moreover, it satisfies an isoperimetric principle stating that graphs with many vertices have a large boundary unless they contain long paths: we show that for graphs with maximal degree $\Delta$ $$ | \partial G| \geq \frac{1}{2\Delta} \frac{|V|}{\mbox{diam}(G)}.$$ For graphs discretizing Euclidean domains, one has $\mbox{diam}(G) \sim |V|^{1/d}$ and recovers the scaling of the classical Euclidean isoperimetric principle.