Perfect codes from PGL(2,5) in Star graphs

Abstract

The Star graph $S_n$ is the Cayley graph of the symmetric group $Sy{m}_n$ with the generating set $\{(1\mbox{ }i): 2\leq i\leq n \}$. Arumugam and Kala proved that $\{\pi \in Sy{m}_n:\pi (1)=1\}$ is a perfect code in $S_n$ for any $n,n\ge 3$. In this note we show that for any $n,n\ge 6$ the Star graph $S_n$ contains a perfect code which is a union of cosets of the embedding of $PGL(2,5)$ into $Sy{m}_6$.

References in corpus (3)

• A note on a Cayley graph of S_n
• $PI$-eigenfunctions of the Star graphs
• Spectra of Cayley graphs