On the automorphism group of a Johnson graph

Abstract

The Johnson graph $J(n,i)$ is defined to the graph whose vertex set is the set of all $i$-element subsets of $\{1,\dots ,n\}$, and two vertices are joined whenever the cardinality of their intersection is equal to $i-1$. In Ramras and Donovan [\emph{SIAM J. Discrete Math}, 25(1): 267-270, 2011], it is conjectured that if $n=2i$, then the automorphism group of the Johnson graph $J(n,i)$ is $S_n\times ⟨T⟩$, where $T$ is the complementation map $A\mapsto \{1,\dots ,n\}\setminus A$. We resolve this conjecture in the affirmative. The proof uses only elementary group theory and is based on an analysis of the clique structure of the graph.