  Let $G$ be a graph on $n$ vertices and $S$ a subset of vertices of $G$; the
boundary of $S$ is the set, $\partial S$, of edges of $G$ connecting $ S $ to
its complement in $G$. The isoperimetric number of $G$, is the minimum of
$\left| \partial S \right|/\left| S \right|$ overall $S \subset V(G)$ of at
most $n/2$ vertices. Let $k \le n$ be positive integers. The Johnson graph is
the graph, $J(n,k)$, whose vertices are all the subsets of size $k$ of
$\{1,\dots,n\}$, two of which are adjacent if their intersection has
cardinality equal to $k-1$. In this paper we show that the asymptotic value of
the isoperimetric number of the Johnson graph $J(n,2)$ is equal to $
(2-\sqrt{2})n$.
