  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.
