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 \( (2-\sqrt{2})n\) .
