The automorphism groups of Johnson graphs revisited

Morteza Mirafzal

Published 2017-02-08Version 1

The Johnson graph $J(n, i)$ is defined to 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) \rtimes Sym(2)$. In this paper we will find these results by different methods. We will prove the conjecture in the affirmative.