A note on the automorphism groups of Johnson graphs

TL;DR

This paper proves that the automorphism group of Johnson graphs $J(n,i)$ is isomorphic to $Sym(n)$ when $n eq 2i$, and to $Sym(n) imes bZ_2$ when $n=2i$, confirming a previous conjecture.

Contribution

The paper provides a new proof confirming the conjecture about the automorphism groups of Johnson graphs, using different methods from prior work.

Findings

Automorphism group of $J(n,i)$ is $Sym(n)$ if $n eq 2i$.

Automorphism group of $J(n,i)$ is $Sym(n) imes bZ_2$ if $n=2i$.

The conjecture by Ramras and Donovan is proven true.

Abstract

The Johnson graph $J(n, i)$ is defined as the graph whose vertex set is the set of all $i$-element subsets of $\{1, . . ., n \}$, and two vertices are adjacent whenever the cardinality of their intersection is equal to $i$-1. In Ramras and Donovan [SIAM J. Discrete Math, 25(1): 267-270, 2011], it is proved that if $ n \neq 2i$, then the automorphism group of $J(n, i)$ is isomorphic with the group $Sym(n)$ and it is conjectured that if $n = 2i$, then the automorphism group of $J(n, i)$ is isomorphic with the group $ Sym(n) \times \mathbb{Z}_2$. In this paper, we will find these results by different methods. We will prove the conjecture in the affirmative.