A Note on the Asymptotic Value of the Isoperimetric Number of $J(n,2)$

Abstract

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$.