The Boundary of a Graph and its Isoperimetric Inequality

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 $Δ$ $$ | \partial G| \geq \frac{1}{2Δ} \frac{|V|}{\mbox{diam}(G)}.$$ For graphs discretizing Euclidean domains, one has $\mbox{diam}(G) \sim |V|^{1/d}$ and recovers the scaling of the classical Euclidean isoperimetric principle.

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